General topology, point-set topology, or analytic toplogy is the study of the invariant properties of objects under continuous transformations, e.g. stretch, twist, scaling. In comparison, geometry is constrained to isometries, e.g. translation, rotation, bending. Topology can be further divided into differential topology, algebraic topology (including combinatorial topology), low-dimensional topology ($\dim M = 2, 3, 4$), etc.

## Concepts

### Topological Structure

Open topology or topological structure $(\mathcal{T}, (\cup_\alpha, \cap))$ is an algebraic system consisting of a subset $\mathcal{T}$ of a power set, which includes the empty set and the underlying set, i.e. $\{\emptyset, X\} \subset \mathcal{T} \subset \mathcal{P}(X)$, and two operations: (1) arbitrary union $\cup_\alpha$ and (2) intersection $\cap$ (such that the system is closed under arbitrary union and finite intersection). Closed topology $(\mathcal{T}^∗, (\cap_\alpha, \cup))$ is the dual concept of open topology, which exchanges union with intersection. Given a topological structure $(\mathcal{T}, (∗_\alpha, \star))$ of either type for a set, the other $(\mathcal{T}^∗, (\star_\alpha, ∗))$ is its dual consisting of the complement elements, i.e. complementation $\complement: \mathcal{T} \mapsto \mathcal{T}^∗$ is an isomorphism between open and closed topologies. Compare topological structure with sigma-algebra $(\Sigma, (\cup_{\mathbb{N}}, \cap_{\mathbb{N}}; \complement))$. A topology is said to be weaker than another topology, and similarly the latter stronger than the former, if the former specifies a coarser topological structure of the underlying set: $\mathcal{T}_1 \subset \mathcal{T}_2$. For a set, the weakest topology $\{\emptyset, X\}$ consists of the empty set and itself; the strongest topology is its power set $\mathcal{P}(X)$, aka the discrete topology.

Topological basis $\mathcal{B}$ for some topology on a set $X$ is a cover of the set that is "pre-closed" in intersection: $\bigcup \mathcal{B} = X$; $\forall x \in B_1 \cap B_2$, $B_1, B_2 \in \mathcal{B}$, $\exists B_3 \in \mathcal{B}$: $x \in B_3 \subset B_1 \cap B_2$. Topology generated by a basis $\mathcal{B}$ is the topology $\mathcal{T(B)}$ consisting of unions of members of arbitrary subsets of the basis: $\mathcal{T(B)} := \{\bigcup \mathcal{C} \mid \mathcal{C} \subset \mathcal{B}\}$. If a class of subsets of a set generates a topology on the set, it is a basis of a topology on the set. The topology generated by a basis is the weakest topology containing the basis: $\mathcal{T(B)} = \cap_{\mathcal{B} \subset \mathcal{T}} \mathcal{T}$. A topology can have many bases, the largest of which is itself.

### Topological Space

Topological space $(X, \mathcal{T})$ is a set $X$ with a topology $\mathcal{T}$. Open subset $A$ of a topological space, or open set in short, is any element of the topology: $A \in \mathcal{T}$. Closed subset $A$ of a topological space, or closed set, is any element of the closed topology, i.e. its complement is an open subset: $A \in \mathcal{T}^∗ = \{X \setminus B \mid B \in \mathcal{T}\}$. For any set endowed with any topological structure, the empty set and the set itself are both open and closed subsets.

Closure (闭包) $\bar{A}$ or $\text{cl}(A)$ of a subset $A$ of a topological space is the smallest closed subset containing $A$: $\bar{A} = \bigcap \{B \mid A \subset B, B \in \mathcal{T}^∗\}$. Interior $\text{Int} A$ of a subset $A$ of a topological space is the biggest open set contained in $A$: $\text{Int} A = \bigcup \{S \mid S \subset A, S \in \mathcal{T}\}$. Exterior $\text{Ext} A$ of a subset $A$ of a topological space is the complement of its closure: $\text{Ext} A = X \setminus \bar{A}$. Boundary $\partial A$ of a subset $A$ of a topological space is the set of points not in neither its interior nor its exterior, or equivalently, the set of points in its closure but not in its interior: $\partial A = X \setminus (\text{Int} A \cup \text{Ext} A) = \bar{A} \setminus \text{Int} A$. A topological space is the disjoint union of the interior, boundary, and exterior of any subset. The closure of a subset is the disjoint union of the interior and the boundary of the subset. For any subset of a topological space, its interior and exterior are open subsets, and its boundary is a closed subset.

Limit point of a subset of a topological space is a point in the space whose punctured neighbourhoods always intersect with the subset. A subset is closed if and only if it contains all its limit points (aka derived set $A'$). The closure of a subset is its union with all its limit points: $\bar{A} = A \cup A'$. Isolated point of a subset of a topological space is a point of the subset with a punctured neighbourhood disjoint from the set. Any point of a subset is either a limit point or an isolated point.

A sequence $(x_n)$ of points in a topological space is eventually in a subset if it has a tail contained in the subset: $\exists N: (x_n)_{n \ge N} \subset A$. A sequence of points in a topological space converges to a point in the space if any neighborhood of the point contains a tail of the sequence: $\forall U(x) \in \mathcal{T}$, $\exists N \in \mathbb{N}$: $(x_n)_{n \ge N} \subset U$. We say $(x_n)$ is a convergent sequence in the space, and $x$ is the limit of the sequence: $\lim_{n \to \infty} x_n = x$. Convergent sequences in a Hausdorff space have unique limits. For a convergent sequence, its limit is its only limit point; any term not equal to its limit is an isolated point. Every subsequence of a convergent sequence in a Hausdorff space converges to the same limit. A sequence of points in a topological space diverge to infinity if every compact subspace contains at most a finite subset of the sequence. In a first countable Hausdorff space, a sequence diverges to infinity if and only if it has no convergent subsequence.

### Countability and Separation Axioms

Countability properties (first and second countability, separability, the Lindelöf property) ensure that a topological space does not have too many open subsets; separation axioms ($(T_i)_{i=0}^4$; Kolmogorov, Hausdorff, regularity, normality) ensure that it has enough open subsets to conform to our spatial intuition.

Neighborhood $U(x)$ of a point $x$ in a topological space is an open set containing the point: $x \in U(x) \in \mathcal{T}$; neighborhood of a subset of a topological space is an open set containing the subset. The term "neighborhood" can be rather misleading: it does not need to be connected, precompact, or "small" (if relative volume makes sense). Neighborhood basis $B_p$ for a topological space $X$ at a point $p$ is a collection of neighborhoods of $p$ such that every neighborhood of $p$ contains some member of $B_p$. First countable topological space is one where every point has a countable neighborhood basis. Every metric space is first countable. Nested neighborhood basis $\{U_i\}$ for $X$ at $p$ is a sequence of neighborhoods of $p$ that are successive subsets and form a neighborhood basis. Every point in a first countable topological space has a nested neighborhood basis. For a first countable topological space, a subset $A$ is: open if and only if every sequence converging to a point in $A$ is eventually in $A$; closed if and only if it contains the limits of every convergent sequence of points in $A$.

Dense subset of a topological space is a subset whose closure is the set: $\bar{A} = X$. Nowhere dense subset of a topological space is a subset whose exterior is dense. Meager subset of a topological space is a subset that can be expressed as a countable union of nowhere dense subsets. Separable space is a topological space containing a countable dense set. Euclidean spaces are separable.

Cover $\mathcal{U}$ of a topological space $X$ is a collection of its subsets whose union is $X$. Open cover is a cover consisting of open subsets. Closed cover is a cover consisting of closed subsets. Subcover $\mathcal{U}'$ of a cover $\mathcal{U}$ of $X$ is a subcollection of $\mathcal{U}$ that still covers $X$. Lindelöf space is a topological space of which every open cover has a countable subcover.

Second countable topological space is one with a countable basis for its topology. A second countable topological space is first countable, separable, and Lindelöf. For metric spaces, second countability, separability, and the Lindelöf property are equivalent.

Kolmogorov space or T_0-space is a topological space where distinct points can be separated by an open set. T_1-space is a topological space where distinct points can be separated by two open sets, one containing either of the points. Hausdorff space or T_2-space is a topological space where distinct points have disjoint neighborhoods. The topology of a Hausdorff space is a Hausdorff topology. Metric spaces and almost all topological spaces encountered in analysis are Hausdorff. Regular space or T_3-space is a Hausdorff space where closed subset and exterior point have disjoint neighborhoods. Completely regular space or Tikhonov space is a Hausdorff space where closed subset and exterior point are functionally separable. Normal space or T_4-space is a Hausdorff space where disjoint closed subsets have disjoint neighborhoods. Urysohn's Lemma: Every normal space is completely regular.

### Connectedness and Compactness

Disconnected topological space $(X, \mathcal{T})$ is one that can be expressed as the union of two disjoint, nonempty, open subsets: $\exists U, V \in \mathcal{T} \setminus \emptyset$: $U \cap V = \emptyset$, $U \cup V = X$. Connected topological space is one that is not disconnected. A topological space is connected if and only if no subset is both open and closed except for the empty set and itself: $\mathcal{T} \cap \mathcal{T}^∗ = \emptyset \cup X$. Component of a topological space $X$ is a maximal nonempty connected subset of $X$. A topological space is partitioned by its components. Components of a topological space are closed subsets. Path in a topological space $X$ from a point $p$ to another point $q$ is a continuous map $f: I \mapsto X$ connecting $p$ and $q$: $f(0) = p, f(1) = q$. Path-connected topological space is one where path exists for any pair of points. If a topological space is path-connected, then it is connected. Locally (path-)connected topological space is one that admits a basis of (path-)connected open subsets: every neighborhood of every point contains a (path-)connected neighborhood of the point. Components of a locally connected topological space are open subsets. If a topological space is locally path-connected, then it is locally connected. A locally path-connected space is connected if and only if it is path-connected.

Compact topological space is one where every open cover has a finite subcover: $\mathcal{U} \subset \mathcal{T}$, $\bigcup \mathcal{U} = X$, then $\exists \{U_i\}_{i=1}^k \subset \mathcal{U}$, $\cup_{i=1}^k U_i = X$. A subspace of a topological space is compact if and only if every cover of the subspace by open subsets of the space has a finite subcover. Note that sometimes a subset of a topological space may be said to have some property of a topological space such as connectedness or compactness, which should be understood as the subspace, i.e. the subset endowed with the subspace topology. Closed subspace of a compact space is compact. Compact subspace of a Hausdorff space is closed. Compact subspace of a metric space is bounded. Compact metric space is complete. A subspace of a Euclidean space is compact if and only if it is closed and bounded.

Limit point compact topological space is one where every infinite subset has a limit point. Compactness implies limit point compactness. Sequentially compact topological space is one where every sequence has a convergent subsequence. Sequentially compactness is also known as the Bolzano-Weierstrass property. For first countable Hausdorff spaces, limit point compactness implies sequential compactness. For second countable topological spaces, sequential compactness implies compactness. For metric spaces and second countable Hausdorff spaces, limit point compactness, sequential compactness, and compactness are equivalent.

Locally compact topological space is one that has a collection of compact subspaces whose interiors cover the space. Precompact or relatively compact subspace $A$ of a topological space $X$ is a subspace whose closure $\bar{A}$ is a compact subspace. For a Hausdorff space, the following are equivalent: it is locally compact; it has a basis of precompact open subsets; each point in it has a precompact neighborhood. A subspace of a locally compact Hausdorff space is again locally compact and Hausdorff, if it is an open or closed subset. Baire space is a topological space such that every countable intersection of dense open subsets is dense. Baire Category Theorem: locally compact Hausdorff spaces and complete metric spaces are Baire spaces. In a Baire space, the complement of a meager subset is dense. Exhaustion $\{K_n\}$ of a noncompact topological space $X$ by compact subspaces is a sequence of compact subspaces each includes in its predecessors in its interior and expands to $X$: $K_n \subset \text{Int}K_{n+1}$, $\bigcup_n K_n = X$. Locally compact, Hausdorff, second countable space admits exhaustion by compact sets.

Locally finite collection of subsets of a topological space is one such that there is an open cover where each set intersects with at most a finite subcollection of the collection. Refinement $\mathcal{B}$ of a cover $\mathcal{A}$ of a topological space $X$ is a cover of $X$ such that each set in $\mathcal{B}$ has a superset in $\mathcal{A}$: $\forall B \in \mathcal{B}, \exists A \in \mathcal{A}: B \subset A$. Open refinement of a cover is a refinement by an open cover. A topological space $X$ has finite topological dimension if there is an integer $k$ such that every open cover has an ("(k+1)-uniformly locally finite") open refinement such that no point lies in more than $k+1$ of the subsets. Topological dimension of such a topological space is the smallest such integer. Paracompact (仿紧) topological space $X$ is one where every open cover admits a locally finite open refinement. Locally compact, Hausdorff, second countable space is paracompact. Paracompact Hausdorff space is normal.

### Derived Topological Spaces

Subspace topology $\mathcal{T}_S$ on a subset $S$ of a topological space $(X, \mathcal{T})$ is the collection of intersections of $S$ with the elements in the topology $\mathcal{T}$: $\mathcal{T}_S = \{S \cap V \mid V \in \mathcal{T}\}$. Subspace topology $\mathcal{T}_S$ is a topology on $S$. Topological Subspace $(S, \mathcal{T}_S)$ of a topological space $(X, \mathcal{T})$ is a topological space consisting of a subset $S$ of $X$ and its subspace topology $\mathcal{T}_S$. Subspace preserves first and second countability, and the Hausdorff property. A special example of subspace is the subset $\mathbb{Q}$ of rational numbers in the space of real numbers $\mathbb{R}$ endowed with the Euclidean topology: it is infinitely disconnected, its components are singletons, but it does not have the discrete topology and no singleton is open, and thus it is not locally connected; it is second countable, Hausdorff, but not locally compact, because the closure of nonempty open subsets are all noncompact (proof not evident); it is not a manifold, of course. Fiber $f^{-1}(y)$ over $y$ of a map $f: X \mapsto Y$ between topological spaces is the subspace that corresponds to the level set of $f$ at $y$.

Product topology $\mathcal{T(B_\times)}$ of topologies $\mathcal{T}_X$ and $\mathcal{T}_Y$ is the topology generated by the basis consisting of the product open sets: $\mathcal{B}_\times = \{U \times V \mid U \in \mathcal{T}_X, V \in \mathcal{T}_Y\}$. Product space $(X \times Y, \mathcal{T(B_\times)})$ of topological spaces $X$ and $Y$ is the topological space consisting of their Cartesian product $X \times Y$ and the product topology $\mathcal{T(B_\times)}$. Product space preserves first and second countability, the Hausdorff property, (path-)connectedness, and compactness.

Disjoint union topology $\mathcal{T}_\sqcup$ of an indexed family $\{\mathcal{T}_\alpha \mid \alpha \in A\}$ of topologies is the topology consisting of the disjoint unions of open sets: $\mathcal{T}_\sqcup = \{\sqcup_\alpha U_\alpha \mid U_\alpha \in \mathcal{T}_\alpha\}$. Disjoint union space $(\sqcup_\alpha X_\alpha, \mathcal{T}_\sqcup)$ of an indexed family $\{(X_\alpha, \mathcal{T}_\alpha) \mid \alpha \in A\}$ of topological spaces is the topological space consisting of the disjoint union $\sqcup_\alpha X_\alpha$ and the disjoint union topology $\mathcal{T}_\sqcup$. Disjoint union space preserves first countability, second countability, and the Hausdorff property.

Quotient topology (商拓扑) $\mathcal{T}_q$ induced by a surjective map $q: X \mapsto Y$ from a topological space $(X, \mathcal{T})$ onto a set $Y$ is the topology consisting of subsets of $Y$ with open preimages: $\mathcal{T}_q = \{U \subset Y \mid q^{-1}(U) \in \mathcal{T}\}$. When the range of a surjective map on a topological space is endowed with the quotient topology, the map is called a quotient map, which is a continuous map. Saturated subset of the domain of a map $f: X \mapsto Y$ between topological spaces is the preimage of some subset of the codomain: $U = f^{-1}(V), V \subset Y$. A surjective continuous map is a quotient map if and only if it takes saturated open/closed subsets to open/closed subsets. Quotient space (商空间) or identification space $(q(X), \mathcal{T}_q)$ of a topological space X by an equivalence relation $\sim$---or its canonical projection q(x) = where is the equivalence class of x w.r.t. $\sim$--- is the topological space consisting of the quotient set $X/\sim$ (set of equivalence classes) and the quotient topology $\mathcal{T}_q$ induced by the canonical projection. Quotient space preserves (path-)connectedness and compactness.

Adjunction space (黏着空间) $X \cup_f Y$ formed by attaching a topological space $Y$ to a topological space $X$ along a continuous map $f: A \mapsto X$ on a closed subspace $A$ of $Y$ is the quotient space of the disjoint union of $X$ and $Y$ by equating each point in $A$ with its value: $X \cup_f Y = q(X \sqcup Y)$, where the canonical projection satisfies $q(x) = \{x\} \sqcup f^{-1}(x), x \in X$ and $q(y) = \{y\}, y \in Y \setminus A$. Note that $X \cup_f Y$ is called "the adjunction space formed by attaching $Y$ to $X$ along $f$", and $f$ is called the attaching map (贴映射). Adjunction space $X \cup_f Y$ is the disjoint union space of the quotient spaces of $X$ and $Y \setminus A$ under the canonical projection: $q(X \sqcup Y) = q(X) \sqcup q(Y \setminus A)$.

### Continuous Map

Continuous map between topological spaces at a point is a map that maps every convergent sequence to the point to a convergent sequence to its value at the point: $\lim_{i \to \infty} x_i = x$ then $\lim_{i \to \infty} f(x_i) = f(x)$; denoted as $\lim_{x_i \to x} f(x_i) = f(x)$. Continuous map $f: X \to Y$ from a topological space $(X, \mathcal{T}_X)$ to a topological space $(Y, \mathcal{T}_Y)$ is a map such that the preimage of any open set is open: $\forall G \in \mathcal{T}_Y, f^{-1}(G) \in \mathcal{T}_X$. Equivalently, a continuous map between topological spaces is a map that is continuous at every point. The set of all continuous maps between two topological spacaes is denoted as $C(X,Y)$ or $C^0(X,Y)$. Continuous real-valued function algebra $C(X)$ on a Hausdorff space is the algebra consisting of the set of all continuous real-valued functions endowed with scalar multiplication, pointwise addition, and pointwise multiplication; i.e. the continuous real-valued function space endowed with pointwise multiplication. Continuous map preserves (path-)connectedness, compactness, the Baire property. Intermediate Value Theorem: The range of a continuous real-valued function on a connected topological space is an interval. Extreme Value Theorem: The range of a continuous real-valued function on a compact topological space has a maximum and a minimum.

Bump function for a subset of a topological space is a continuous extension of the indicator function on the subset: $f \in C(X, I)$, $f|_A = 1$. Functionally separable subsets in a topological space are two subsets such that there is a bump function for one subset that equals zero on the other: $\exists f \in C(X, I)$: $f|_A = 0$, $f|_B = 1$. A set of continuous functions on a Hausdorff space separates points in the space if every pair of distinct points can be separated by a function in the set: $x, y \in X$, $x \ne y$ then $\exists f \in C(X)$: $f(x) \ne f(y)$. If a set of continuous functions on a Hausdorff space separates points in the space, then the preimages of open subsets on the real line generates the topology of the space: $\mathcal{T}\{f^{-1}(U) : f \in A, U \in \mathcal{T}_{\mathbb{R}}\} = \mathcal{T}_X$. Support $\text{supp} f$ of a continuous real-valued function on a topological space is the closure of the subset where the function is not zero: $\text{supp} f = \overline{X \setminus f^{-1}(0)}$. Every closed subset of a normal topological space has a bump function supported in an arbitrary neighborhood of the subset.

Exhaustion function for a topological space $(X, \mathcal{T})$ is a continuous real-valued function whose sublevel sets $f^{-1}(-\infty, c]$ are compact.

Partition of unity $\{\psi_\alpha \mid \alpha \in A\}$ on a topological space $X$ subordinate to an indexed open cover $\mathcal{X} = \{X_\alpha \mid \alpha \in A\}$ of $X$ is an indexed family of continuous non-negative functions suported within the corresponding element of the cover that add up to one in locally finite sums: $\sum_\alpha \psi_\alpha (x) = 1$; $\text{supp}\psi_\alpha \subset X_\alpha$; $\forall p \in X, \exists U(p) \in \mathcal{T}$: $\exists B \subset A, |B| < \infty$, $\sum_{\alpha \in B} \psi_\alpha (x) = 1$. For a paracompact Hausdorff space, every open cover admits a subordinate partition of unity.

Topological isomorphism (拓扑同构; isomorphisms in the category of topological spaces), or homeomorphism (同胚), is a continuous map between two topological spaces that has a continuous inverse map. Two topological spaces are topologically equivalent, or homeomorphic, if there is a homeomorphism between them. Topological invariant of a topologically equivalent class of topological spaces is a property common to all these spaces, e.g. number of components, (path-)connectedness, compactness, the Baire property, etc. To prove that two topological spaces are homeomorphic is usually straightforward, i.e. through definition, by constructing a homeomorphism; To prove that two topological spaces are not homeomorphic is usually by showing that they have different topological invariants.

Topological embedding (嵌入) $f: X \to Y$ of a topological space into another topological space is an injective continuous map that yields a homeomorphism between its domain and its range with the subspace topology. An embedding is a representation of a topological space in another topological space, which preserves its connectivity or algebraic properties. Reach $r$ of a topological space embedded in a Euclidean space is the largest real number such that any point with a distance less than the number from the space has a unique projection on the space. Reach can be zero for non-smooth manifolds, such as the graph of $|x|$, aka "corner" (it is the boundary of the epigraph of $|x|$, where the origin is the corner point).

Section (截面) $\sigma$ of a surjective continuous map $\pi: M \mapsto N$ is a continuous map $\sigma: N \mapsto M$ whose composition with $\pi$ is the identity map on $N$: $\pi \circ \sigma = \text{Id}_N$. Equivalently, a section of a surjective continuous map is a continuous right inverse of the map. Local section of a continuous map is a section on an open subset of the codomain: $\sigma: U \mapsto M$, $U \subset N$, $\pi \circ \sigma = \text{Id}_U$.

Evenly covered open subset of the codomain of a continuous map is one whose preimage is a disjoint union of connected open subsets of the domain, the restriction of the map to each is a homeomorphism to the subset. Every evenly covered open subset is connected. Sheet of the covering over an evenly covered open set is a component of the preimage of the set. Covering map $q: E \mapsto X$ is a surjective continuous map on a connected and locally path-connected topological space such that the codomain has an open cover consisting of evenly covered sets. For a covering map, the domain is called the covering space of the codomain, and the codomain is called the base of the covering. Lift $\tilde\phi: Y \mapsto E$ of a continuous map $\phi: Y \mapsto X$, given a covering map $q: E \mapsto X$ onto its codomain, is a continuous map that equals the original map when followed by the covering map: $q \circ \tilde\phi = \phi$. The lift of a continuous map from a connected topological space, given a covering map, is uniquely determined by its value at any one point: $\exists p \in Y$, $\tilde\phi_1(p) = \tilde\phi_2(p)$ then $\tilde\phi_1 = \tilde\phi_2$. Lifting problem for covering maps is the problem of deciding whether a continuous map admits a lift to a covering space of its codomain.

### Closed Map and Open Map

Closed map between topological spaces is a map that maps closed subsets to closed subsets: $A \mathcal{T}_X^∗$ then $f(A) \in \mathcal{T}_Y^∗$. Closed Map Lemma: A continuous map from a compact space to a Hausdorff space is a closed map; in addition, if it is surjective, it is a quotient map; if it is injective, it is a topological embedding; if it is bijective, it is a homeomorphism.

Proper map between topological spaces is a map such that the preimage of a compact subset is compact. Proper map preserves divergence to infinity. Compactly generated topological space $X$ is one such that if the intersection of a subset $A$ with any compact subset $K$ is closed in $K$ then $A$ is closed in $X$. A proper continuous map to a compactly generated Hausdorff space is a closed map.

Open map between topological spaces is a map that maps open subsets to open subsets. For maps between topological spaces, openness, closedness, and continuity are independent.

## Manifold (General Topology)

Manifold builds on a Euclidean space to obtain new spaces that are topologically distinct to the Euclidean space (or its regular domains). Manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

Topological manifold without boundary is a topological space $(M, \mathcal{T})$ that is Hausdorff, second-countable, and locally Euclidean of a certain dimension $n$, i.e. has an open cover where each set is homeomorphic to (an open subset/ball of) the n-dimensional Euclidean space $\mathbb{R}^n$. The definitions of manifold are equivalent whether using open subset, open ball, or the Euclidean space itself. Here $n$ is called the dimension $\dim M$ of the manifold. A topological manifold of dimension $n$ is also known as a topological n-manifold. Open subsets of the Euclidean n-space are n-manifolds. n-manifold topology is the topology of an n-manifold.

Closed n-dimensional upper half-space $\mathbb{H}^n$ is the subspace of the Euclidean n-space with non-negative n-th component: $\mathbb{H}^n = \{x \in \mathbb{R}^n \mid x_n \ge 0\}$. n-manifold with boundary is a topological space that is Hausdorff, second countable, and has an open cover where some sets are homeomorphic to (an open subset of) the Euclidean n-space and the others are homeomorphic to (an open subset of) $\mathbb{H}^n$. Most propositions true for boundaryless manifolds are also true for manifolds with boundary, perhaps with minor modifications; unless explicitly stated, manifolds in this article refers to those with and without bounary.

Dimension is a topological invariant of a manifold. An n-manifold has topological dimension $n$.

Manifold has many nice properties for a topological space: manifold is locally path-connected, locally compact, paracompact, and normal; any subset of a manifold is compactly generated; the fundamental group of a manifold is countable; (smooth) n-manifold admits a (smooth) embedding into the Euclidean 2n-space. Compact manifolds can be embedded into closed and bounded subsets of some Euclidean space.

Any closed subset $A$ of a manifold $M$ can be the zero set of a continuous non-negative function $f$ on the manifold: $f^{-1}(0) = A$. Strengthening Urysohn’s Lemma, any closed subset $B$ disjoint to $A$ can be made the preimage of one, while other points take values in between: $f^{-1}(1) = B$, $f(M) = I$. Manifold admits a positive exhaustion function.

### Examples

The empty set is a manifold of any dimension. Countable discrete spaces $(N, 2^N)$, $N \subset \mathbb{N}$, are all the 0-manifolds. A rubber band is a 1-manifold; a broken rubber band is a 1-manifold with 0-dimensional boundary. The unit circle $\mathbb{S}^1$ is a 1-manifold; the unit interval $I = [0, 1]$ is a 1-manifold with 0-dimensional boundary. A balance ball is a 2-manifold; half a balance ball is a 2-manifold with 1-dimensional boundary. The unit sphere $\mathbb{S}^2$ is a 2-manifold; the closed unit disk $\bar{\mathbb{B}}^2$ is a 2-manifold with 1-dimensional boundary. Cylinder $\mathbb{S}^1 \times I$ and the Möbius band are 2-manifolds with 1-dimensional boundaries. The torus $\mathbb{T}^2$ and the Klein bottle $K$ are 2-manifolds. The projective plane $\mathbb{P}^2$ is a 2-manifold. A medicine ball is a 3-manifold with 2-dimensional boundary.

Real projective space $\mathbb{P}^n$ (or $\mathbb{RP}^n$) of dimension $n$ is the quotient space of $\mathbb{R}^{n+1} \setminus \{0\}$ by the projection map $q(x) = \{ax \mid a \ne 0\}$ equating points up to nonzero scaling. Note that $\mathbb{P}^n$ is not homeomorphic with $\mathbb{S}^n$ because opposite/antipodal points are equivalent. The n-dimensional real projective space $\mathbb{P}^n$ is an n-manifold, and it is homeomorphic to the n-sphere via the covering map $\text{span}: \mathbb{S}^n \mapsto \mathbb{P}^n$ (for n=1, trivial proof; for n>1, [@Lee2011, Cor 11.33]). Complex projective space $\mathbb{CP}^n$ is the quotient space of $\mathbb{C}^{n+1} \setminus \{0\}$ by equating points up to nonzero scaling. The n-dimensional complex projective space $\mathbb{CP}^n$ is a 2n-manifold.

The state space of a dynamical system is often considered a manifold (literally, the set of all possible values of a variable with certain constraints), which can be much more complex than a Euclidean space due to conservation laws or other constraints. The dimension of the manifold corresponds to the degrees of freedom of the system, where the points are specified by generalized coordinates. (The configuration space of double pendulum is a 2-torus.) Applications: symplectic manifold for analytical mechanics (Lagrangian, Hamiltonian) [@Arnold1989]; Lorentzian 4-manifold for general relativity; complex manifold for complex analysis.

### Charts

Local coordinate chart $\phi: U \to \mathbb{R}^n$ of an n-manifold or interior chart of an n-manifold with boundary is a homeomorphism from an open subset of the manifold to an open subset of $\mathbb{R}^n$. Boundary chart $\phi: U \to \mathbb{H}^n$ of an n-manifold with boundary is a homeomorphism from an open subset of the manifold to an open subset of $\mathbb{H}^n$ that includes a part of the boundary of the half-space. Interior point of a manifold is a point in the domain of some interior chart. Boundary point of a manifold with boundary is a point in the domain of a boundary chart that sends it to the boundary of the half-space. Interior $\text{Int} M$ of a manifold is the set of all its interior points. The interior of an n-manifold is an open subset of the manifold and an n-manifold without boundary. Boundary $\partial M$ of a manifold with boundary is the set of all its boundary points. The boundary of an n-manifold with boundary is a closed subset of the manifold and an (n-1)-manifold without boundary. Any manifold is the disjoint union of its interior and boundary. The boundary of a manifold as a topological space and as a subset of a topological space are independent concepts.

Local coordinates $(x_i)_{i=1}^n$ on the domain of a local coordinate chart are the component functions of the chart: $\phi = (x_i)_{i=1}^n$. Local coordinate representation $\hat{p}$ of a point $p$ in the domain of a chart is the tuple of local coordinate values at the point: $\hat{p} = \phi(p)$. Atlas $\mathcal{A}$ of a manifold $M$ is a class of coordinate charts whose domains cover $M$. Transition map from a coordinate chart $\phi_1$ to another coordinate chart $\phi_2$ of the same n-manifold, whose domains $U_1$ and $U_2$ overlap, is the composite map $\phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2) \to \mathbb{R}^n$. Note that transition maps are functions between subsets of $\mathbb{R}^n$.

Coordinate domain $U$ is the domain of a local coordinate chart. A manifold admits a finite atlas; the coordinate domains are disconnected in general. Coordinate ball $B$ of an n-manifold is an open subset of the manifold that is homeomorphic to an open ball $B_r(0)$ in $\mathbb{R}^n$. Coordinate half-ball of an n-manifold with boundary is an open subset of the manifold that is homeomorphic to an open half-ball $B_r(0) \cap \mathbb{H}^n$ in $\mathbb{H}^n$. Regular coordinate ball of an n-manifold is one such that there is a chart on a larger coordinate ball that maps it to an open ball and maps its closure to the closed ball: $\phi: B' \mapsto B_{r'}(x)$, $\phi(B) = B_r(x)$, $\phi(\bar{B}) = \bar{B}_r(x)$. A manifold has a countable basis of regular coordinate balls. The complement $M \setminus B$ of a regular coordinate ball $B$ in an n-manifold $M$ without boundary is an n-manifold with boundary and the boundary is homeomorphic to the (n-1)-sphere $\mathbb{S}^{n-1}$.

Global coordinate chart is a coordinate chart on the entire manifold. Each point of a manifold is in the domain of some local coordinate chart, but the manifold may not have a global coordinate chart. In fact, an n-manifold has a global coordinate chart if and only if it is homeomorphic to an open subset or a regular domain of $\mathbb{R}^n$. For example, angular coordinate is a chart of a circle, but not a global homeomorphism; in the same sense, geographical coordinates is a local chart of a sphere.

Table: Abstract Object vs Representations

Abstract Object Representation Reference Invariants
vector space Cartesian power of field basis dimemsion, field
vector tuple basis
linear operator matrix two bases singular value-space pairs
tensor array n bases
linear transformation square matrix basis eigenvalue-eigenspace pairs
inner product space Euclidean space basis length, angle
orthogonal linear trans normal matrix - orthogonal eigen-basis
linear isometry unitary matrix - length, angle
self-adjoint Hermitian matrix - real eigenspaces
scaling positive definite matrix - eigen-direction
strongly connected graph irreducible nonnegative m - max eigenvalue-eigenvector
manifold coordinate balls atlas topological invariants
point coordinate chart
continuous map continuous function chart
smooth manifold embedded submanifold embedding smooth invariants
tangent vector tuple chart
differential Jacobian matrix charts
Riemannian manifold geometric invariants
isometry unitary Jacobian matrix charts distance, volume, curvature

Manifolds are abstract objects that do not come with any predetermined choice of coordinates, and unlike subsets of a Euclidean space, their points often do not have canonical representation. Reversely, a manifold can be identified as a subset of a higher-dimensional Euclidean space, and then an atlas of charts covering this subset is constructed. Manifolds can be also constructed by surgery, i.e. specifying an atlas which is itself defined by transition maps.

Objects defined globally on a manifold are also abstract, and thus shall not depend on a particular choice of coordinates. Invariant definition of an abstract object is a definition that is representation-independent. An abstract object on a manifold can be defined invariantly, if possible; one can also give it a coordinate-dependent definition and then prove that the definition gives the same results in any coordinate chart. For example, a manifold may be endowed with more structures besides a nice topology, such as smooth structure (see Smooth Manifold / Differential Topology) and geometric structure (see Differential Geometry). Such structures are first defined on each chart separately, and if all the transition maps are compatible with the structure, it transfers to the manifold.

### Derived Manifolds

Subspace of a manifold may not be a manifold; for example, cross lines $y^2 - x^2 = 0$, opposite cones $y^2 - x^2 \ge 0$. Recall that the graph $\Gamma(f)$ of a map $f: X \mapsto Y$ is the subset of $X \times Y$ defined as $\Gamma(f) = \{(x, f(x)) \mid x \in X\}$. The graph of any continuous function from any open subset of $\mathbb{R}^n$, endowed with the subspace topology, is an n-manifold homeomorphic to the domain: $\forall U \in \mathcal{T}(\mathbb{R}^n)$, $\forall f \in C^0(U, \mathbb{R}^k)$, $\Phi_f: U \mapsto \Gamma(f)$, $\Phi_f(x) = (x, f(x))$ is a homeomorphism. Projection onto the first $n$ components is a global coordinate chart of the manifold, known as the graph coordinates: $\phi = (\pi_i)_{i=1}^{n}|_{\Gamma(f)}$.

The product space $M \times N$ of an m-manifold and an n-manifold, both without boundary, is an (m+n)-manifold without boundary, which may be call a product manifold. n-torus is the n-th power of 1-spheres: $\mathbb{T}^n = (\mathbb{S}^1)^n$; which is often considered as the subset of the Euclidean 2n-space defined by $\forall i \in \{i\}_{i=1}^n$, $(x^{2i-1})^2 + (x^{2i})^2 = 1$. A finite cylinder $\mathbb{S}^1 \times I$.

Disjoint union of n-manifolds is an n-manifold. Disjoint union of manifolds of different dimensions are not manifolds.

Quotient space of a manifold may be a manifold. For example, closed unit disk $\bar{\mathbb{B}}^2$ to sphere $\mathbb{S}^2$. Polygonal region is a compact subset of $\mathbb{R}^2$ whose boundary is a polygon. The quotient space of a disjoint union of polygonal regions that identifies pairs of edges in affine homeomorphisms (affine maps that identify the two pairs of vertices in either order) is a compact 2-manifold. For example, square $I^2$ to cylinder $\mathbb{S}^1 \times I$ by $(0, t) \sim (1, t)$; square to Möbius band by $(0, t) \sim (1, 1-t)$; cylinder $\mathbb{S}^1 \times I$ to torus $\mathbb{T}^2$ by $(t, 0) \sim (t, 1)$; cylinder to Klein bottle $K$ by $(t, 0) \sim (1-t, 1)$.

If $M$ and $N$ are n-manifolds with boundary and $h: \partial N \mapsto \partial M$ is a homeomorphism between their boundaries, the adjunction space $M \cup_h N$ (formed by attaching $N$ to $M$ along $h$) is called "an adjunction space formed by attaching $M$ and $N$ together along their boundaries", which is an n-manifold without boundary, referred to as an "adjunction manifold" hereafter. The original manifolds can be embedded into an adjunction manifold as closed subsets which cover the adjunction manifold and intersect at the attached boundary. Double $D(M)$ of a manifold $M$ with boundary is the adjunction manifold formed by attaching two copies of $M$ together along the identity map of its boundary: $D(M) = M \cup_h M$, $h = \text{Id}_{\partial M}$. Connected sum $M_1 \# M_2$ of connected n-manifolds $M_i$, $i = 1, 2$, is a connected adjunction n-manifold formed by cutting out a regular coordinate ball $B_i$ on each and attaching the remainders $M'_i = M_i \setminus B_i$ together along their boundaries: $M_1 \# M_2 = M'_1 \cup_f M'_2$, $f: \partial M'_2 \mapsto \partial M'_1$ is a homeomorphism. The Klein bottle is homeomorphic to the connected sum of two projected planes. The connected sums of two given n-manifolds are in at most two homeomorphic classes which differ only in orientation.

### Orientation

Orientable compact surface is one that admits an oriented presentation. The Möbius band is not orientable. A compact connected surface is orientable if and only if it is homeomorphic to the sphere or a connected sum of one or more tori, or equivalently, it does not contain a subset homeomorphic to the Möbius band. Orientability of a compact surface is a topological invariant. Orientable surface of genus 0 is the sphere. Orientable surface of genus n is the connected sum of $n$ tori. Nonorientable surface of genus n is the connected sum of $n$ projective planes. The genus (亏格) of a surface can be recovered from the fundamental group, so it is a topological invariant.

Orientable triangulated manifolds can also be defined...

### Classification by Model Manifolds and Topological Invariants

Classification of 1-manifolds. Every connected 1-manifold is homeomorphic to exactly one of the following: (1) the circle $\mathbb{S}^1$: compact, no boundary; (2) the real line $\mathbb{R}$: noncompact, no boundary; (3) the unit interval $[0, 1]$: compact, with boundary; (4) closed half line $[0, \infty)$: noncompact, with boundary. Although compact 1-manifolds are homeomorphic to the circle, their embeddings may not be isotopic, e.g. nontrivial knots vs the unknot.

Classification of 2-manifolds (aka surfaces). Every compact, connected 2-manifold without boundary is homeomorphic to exactly one of the following: (1) The sphere $\mathbb{S}^2$; (2) A connected sum of one or more copies of the torus $\mathbb{T}^2$; (3) A connected sum of one or more copies of the projective plane $\mathbb{P}^2$.

Poincaré conjecture [@Poincare1904]: Simply connected, compact 3-manifolds are homeomorphic to the 3-sphere $\mathbb{S}^3$. Stephen Smale proved analogous cases for $n≥5$ [@Smale1961]; Michael Freedman proved the analogous case for $n=4$ [@Freedman1982]; Grigori Perelman completed Richard Hamilton's program which uses Ricci flow to prove the existence of geometric decompositions, and thus proved the Thurston geometrization conjecture, which implies the Poincaré conjecture [@Perelman2006].

Thurston geometrization conjecture [@Thurston1982]: Every compact, orientable 3-manifold can be expressed as a connected sum of compact manifolds, each of which either admits a Riemannian covering by a homogeneous Riemannian manifold or can be cut along embedded tori so that each piece admits a finite-volume locally homogeneous Riemannian metric.

No program can decide whether two smooth n-manifolds, $n≥4$, are diffeomorphic. [@Markov1958] It is generally undecidable whether two n-manifolds, $n>4$, are homeomorphic.

## Smooth Manifold (Differential Topology)

Differential topology is the study of the invariant properties of smooth manifolds under smooth deformations (diffeomorphisms). Smoothness is the notion for a topological space to have (co-)tangent spaces that vary continuously, and sucessively higher order differentials that also vary continuously. The smoothness condition, i.e. infinite continuous differentiability, can often be relaxed in practice, because continuous maps (and finitely continuously differentiable maps) can be uniformly approximated by smooth maps. Smooth manifolds admits differentiation and integration, see Calculus on Manifolds.

Smoothly compatible coordinate charts of a manifold are two charts $\phi_1, \phi_2$ such that either their domains overlap and their transition maps $\phi_j \circ \phi_i^{-1}$ are smooth functions, or their domains are disjoint. Smooth atlas of a manifold is an atlas where any two charts are smoothly compatible. Smooth structure, C^∞ structure, or maximal smooth atlas of a manifold is a smooth atlas that is not properly contained in any larger smooth atlas, i.e. any chart that is smoothly compatible with every chart in the atlas is already in it. Smooth chart is any chart in a smooth structure. Smooth structure determined by a smooth atlas of a manifold is the unique maximal smooth atlas $\bar{\mathcal{A}}$ containing that smooth atlas. If a manifold can be covered by a single chart $\phi$, then $\phi$ forms a singleton atlas $\mathcal{A} = \{\phi\}$, which is a smooth atlas and thus determines a smooth structure on the manifold. Two smooth atlases for a manifold determine the same smooth structure if and only if their union is a smooth atlas.

Smooth manifold $(M, \mathcal{T}, \mathcal{A})$ (sometimes called "differentiable manifold") is a manifold $M$ endowed with a smooth structure $\mathcal{A}$. The graph of any smooth function from any open subset of $\mathbb{R}^n$, endowed with the subspace topology and the smooth structure determined by the smooth atlas consisting of the graph coordinate chart, is a smooth n-manifold diffeomorphic to the domain. (See Regular Level Set Theorem for a generalization.) Smooth manifold structure $(\mathcal{T}, \mathcal{A})$ is a manifold topology together with a smooth structure.

Analogous to smooth structure, C^k structure, real-analytic structure or C^ω structure, and complex-analytic structure are defined such that the transition maps between overlapping coordinate charts are, respectively, $C^k$ (weaker than smooth) and real- and complex-analytic (stronger than smooth). Manifold endowed with such a structure is called C^k manifold, real-analytic manifold, and complex manifold, respectively. Note that $C^0$ manifold is equivalent to topological manifold.

Smooth structure of a manifold with boundary is a smooth structure containing boundary charts. Smooth manifold with boundary is a manifold with boundary endowed with a smooth structure. Smooth chart for a manifold with boundary is any chart in its smooth structure. Regular coordinate half-ball of an n-manifold with boundary is one such that there is a smooth chart on a larger coordinate half-ball that maps it to an open half-ball and maps its closure to the closed half-ball: $\phi: B' \mapsto B_{r'}(0) \cap \mathbb{H}^n$, $\phi(B) = B_r(0) \cap \mathbb{H}^n$, $\phi(\bar{B}) = \bar{B}_r(0) \cap \mathbb{H}^n$.

For the nonnegative subspace $\bar{\mathbb{R}}_+^n$ of the Euclidean n-space, its boundary point is a point where at least one coordinate vanish; its corner point, if $n \ge 2$, is a point where at least two coordinates vanish. The number of vanished coordinates can be considered as the "type" of a boundary (or corner) point. Chart with corners $\phi: U \to \bar{\mathbb{R}}_+^n$ for an n-manifold with boundary is a coordinate chart to an open subset of $\bar{\mathbb{R}}_+^n$. A chart with corners may contain any type of boundary points in its range, or none at all. Charts with corners containing different types of boundary points cannot be smoothly compatible. Smooth structure with corners of a manifold with boundary is a smooth structure consisting of charts with corners. Smooth manifold with corners is a manifold with boundary endowed with a smooth structure with corners. Smooth chart with corners is any chart with corners in a smooth structure with corners. Invariance of Corner Points: If a point in a smooth manifold with corners is mapped to a corner point of the nonnegative subspace for some smooth chart with corners, then it is mapped to a corner point (of the same type) in all smooth charts with corners containing it. Boundary point of a smooth manifold with corners is the preimage of a boundary point of the nonnegative subspace in a smooth chart with corners. Corner point of a smooth manifold with corners is the preimage of a corner point of the nonnegative subspace in a smooth chart with corners. A smooth manifold with corners is a smooth manifold with or without boundary if and only if it has no corner points. The boundary of a smooth manifold with corners is in general not a smooth manifold with corners; e.g. $\bar{\mathbb{R}}_+^n$ as a submanifold of $\mathbb{R}^n$ is a smooth n-manifold with corners, but its boundary is merely a finite union of smooth (n-1)-manifolds with corners. Most constructs for smooth manifolds with or without boundary can also be defined on smooth manifolds with corners in the same way, using smooth charts with corners in place of smooth boundary charts.

### Derived Smooth Manifold

Subspace smooth structure $\mathcal{A}_S$ on a subspace $S$ of a smooth manifold $(M, \mathcal{A})$ is the collection of smooth charts restricted to $S$: $\mathcal{A}_S = \{\phi|_S: \phi \in \mathcal{A}\}$.

Product smooth structure is the smooth structure determined by the smooth atlas consisting of products $\phi_1 \times \phi_2$ of smooth charts of component manifolds. The product manifold $M \times N$ of a smooth manifold without boundary and a smooth manifold with boundary is a smooth manifold with boundary, and the boundary is the Cartesian product of the former and the boundary of the latter: $\partial (M \times N) = M \times \partial N$.

Collar neighborhood of the boundary of a smooth manifold $M$ is a neighborhood of the boundary and is the image of a smooth embedding $F: [0,1) \times \partial M \mapsto M$ that restricts to the identification on the boundary: $F(0, p) = p$. Collar neighborhood exists. If $M$ and $N$ are smooth n-manifolds with boundary and $h: \partial N \mapsto \partial M$ is a diffeomorphism between their boundaries, the adjunction manifold $M \cup_h N$ endowed with a smooth structure such that the original manifolds are each diffeomorphic to a regular domain in the adjunction manifold which cover the adjunction manifold and intersect at their boundaries, is a smooth n-manifold without boundary, referred to as a "smooth adjunction manifold". Smooth connected sum of connected smooth n-manifolds is a connected smooth adjunction n-manifold formed just like connected sum but by a diffeomorphism.

Smooth Manifold Chart Lemma: A set $X$ can be endowed with an n-manifold topology and a smooth structure, defined by $\mathcal{T}_\phi = \mathcal{T}(\{ \phi_\alpha^{-1}(V) : V \in \mathcal{T}_{\mathbb{R}^n}, \alpha \in A\})$ and $\mathcal{A}_\phi = \overline{\{\phi_\alpha\}_\alpha}$, given a collection of maps $\phi_\alpha: U_\alpha \mapsto \mathbb{R}^n$ that satisfy the following: (1, 2) homeomorphism: each map is an injection to an open set, and maps the intersection of its domain and another to an open set; (3) smooth compatibility: each composite map $\phi_\beta \circ \phi_\alpha^{-1}$ is smooth; (4) second countablity: the domains of a countable subcollection cover $X$; (5) Hausdorff property: distinct points not in one domain are in disjoint domains. Grassmann manifold $G_k(V)$ is the set of all k-dimensional linear subspaces of an n-dimensional real vector space $V$, endowed with certain smooth structure, is a smooth k(n-k)-manifold. If $V$ is the Euclidean space $\mathbb{R}^n$, Grassmann manifold is denoted as $G_{k,n}$. Projective spaces are special cases of Grassmann manifolds: $G_{1,n+1} = \mathbb{P}^n$.

### Smooth Map

Local coordinate representation $\hat{F}: \phi_1(U_1) \mapsto \phi_2(U_2)$ of a map $F: M_1 \mapsto M_2$ between smooth manifolds w.r.t. smooth charts $\phi_1$ and $\phi_2$ on domains $U_1$ and $U_2$, $F(U_1) \subset U_2$, is the map defined by $\hat{F} = \phi_2 \circ F \circ \phi_1^{-1}$. Smooth map $F: M_1 \mapsto M_2$ between smooth manifolds is a map with smooth local coordinate representations everywhere. The set of all smooth maps from $M$ to $N$ is denoted $C^\infty(M, N)$. The set of all smooth real-valued functions on a manifold $M$ is denoted $C^\infty(M)$, which is a vector space over field $\mathbb{R}$.

A smooth map can be constructed from smooth maps on subsets of the domain by either gluing or blending them together. Gluing Lemma for Smooth Maps: Given smooth maps $F_\alpha: U_\alpha \mapsto N$ on open submanifolds of a smooth manifold $M$ that cover the manifold and the maps are identical on overlapping domains, then there is a unique smooth map $F: M \mapsto N$ matching the maps on their domains. Smooth partition of unity on a smooth manifold is a partition of unity consisting of smooth functions. Smooth partition of unity exists for any smooth manifold $M$ and any indexed open cover of $M$. Any smooth function $f: A \mapsto \mathbb{R}^k$ on a closed subset $A$ of a smooth manifold $M$ can be extended to a smooth function $\tilde{f}$ on $M$ that vanishes on any open subset $U$ containing $A$. Smooth manifold admits a smooth positive exhaustion function. Closed subset $A$ of a smooth manifold $M$ can be the zero set of a smooth non-negative function $f$ on the manifold: $f^{-1}(0) = A$.

Smooth isomorphism or diffeomorphism (微分同胚) between smooth manifolds is a smooth map with a smooth inverse. Two smooth manifolds are smoothly equivalent or diffeomorphic, denoted as $\approx$, if there is a diffeomorphism between them. Note that although the real projective n-space and the n-sphere are homeomorphic, they are diffeomorphic if and only if n = 1. The complex projective 1-space and the 2-sphere are diffeomorphic. The complex projective n-space and $\mathbb{S}^{2n+1} / \mathbb{S}^1$ are diffeomorphic.

Smooth invariant or diffeomorphic invariant of a smoothly equivalent class of smooth manifolds is a property common to all these manifolds. Diffeomorphism is a subclass of homeomorphism, and smooth invariants are not necessarily topological invariants.

### Tangent Bundle

Derivation (导数) $v: C^\infty(M) \mapsto \mathbb{R}$ at a point $p$ of a smooth manifold $M$ is a functional on smooth real-valued functions that is linear over $\mathbb{R}$ and satisfies the product rule at $p$: $\forall a, b \in \mathbb{R}$, $\forall f, g \in C^\infty(M)$, $v (a f + b g) = a (v f) + b (v g)$, $v (f g) = (v f) g + f (v g)$. Tangent space $T_p M$ at a point $p$ on a smooth manifold $M$ is the set of all derivations at $p$. Tangent space $T_p M$ is a vector space of dimension $n = \dim M$, and derivations at $p$ are thus also called tangent vectors at $p$. The canonical isomorphism $F: V \mapsto T_a V$ from a vector space to its tangent space at a point is writen as $F(v) = D_v|_a$, defined by $D_v|_a f = \frac{d}{dt}\Bigg{|}_{t=0} f(a+tv)$. In particular, $\mathbb{R}^n \cong T_a \mathbb{R}^n$ by such identification. The tangent space to a product manifold can be identified with the direct sum of the tangent spaces of the component manifolds: $T_{(p,q)} (M \times N) \cong T_p M \oplus T_q N$.

Coordinate vector (坐标切向量) $\partial / \partial x^i |_p$ at a point $p$ on a smooth manifold $M$ associated with a smooth chart containing $p$ is the tangent vector at $p$ that equals the i-th partial derivative of the coordinate representation of any function at the coordinate representation of the point: $\forall f \in C^\infty(M)$, $\frac{\partial}{\partial x^i} \bigg{|}_p f = \frac{\partial \hat f}{\partial x^i} (\hat p)$. Coordinate basis $(\partial / \partial x^i |_p)_{i=1}^n$ for a tangent space $T_p M$ associated with a smooth chart containing $p$ is the n-tuple of the coordinate vectors at $p$ associated with the chart. A coordinate basis for a tangent space is an algebraic basis. Component function $v^i$ of a tangent vector $v$ w.r.t. a coordinate basis is the value of $v$ applied to the i-th coordinate function: $v^i = v(x^i)$. Local coordinate representation $\hat{v}$ of a tangent vector $v$ w.r.t. a coordinate basis is the n-tuple of its components: $\hat{v} = (v^i)_{i=1}^n$. Tangent vector can be written uniquely as a linear combination of a coordinate basis and its local coordinate representation wherein: $v = v^i \frac{\partial}{\partial x^i} \bigg{|}_p$.

Pointwise differential $d F_p$ or tangent map $T_p F$ of ---or pushforward (前推) $F_∗$ of tangent vectors by---a smooth map $F \in C^\infty(M, N)$ at a point $p \in M$ is a linear operator $d F_p \in \mathcal{L}(T_p M, T_{F(p)} N)$ that takes a tangent vector $v$ at the point to the tangent vector at its value that equals $v$ acting on the composition of smooth real-valued functions with the map: $\forall f \in C^\infty(N)$, $d F_p(v)(f) = v(f \circ F)$. Local coordinate representation $\widehat{d F_p}$ of a differential $d F_p$ w.r.t. the coordinate bases for $T_p M$ and $T_{F(p)} N$ associated with smooth charts $\phi$ and $\psi$ is the Jacobian matrix of the coordinate representation of $F$ w.r.t. the charts: $\widehat{d F_p} = \left[\frac{\partial \hat{F}^j}{\partial x^i} (\hat p)\right]$. (Note that the columns and rows are the opposite of the convention in Analysis.) The pushforward of coordinate vectors by a smooth map can be written as: $d F_p \left(\frac{\partial}{\partial x^i} \bigg{|}_p \right) = \frac{\partial \hat{F}^j}{\partial x^i} (\hat p) \frac{\partial}{\partial y^j} \bigg{|}_{F(p)}$.

Tangent bundle (切丛) $T M$ of a smooth manifold $M$ is the disjoint union of the tangent spaces at all points of $M$: $T M = \sqcup_{p \in M} T_p M$. Elements of a tangent bundle are often written as $(p, v)$, point before tangent vector. Natural projection map $\pi: T M \mapsto M$ of a tangent bundle sends each indexed tangent vector to the point of tangent: $\pi(p, v) = p$. Natural coordinates $(\hat{p}, \hat{v})$ of an element $(p, v)$ of a tangent bundle associated with a smooth chart is the tuple of the local coordinate representation of $p$ w.r.t. the chart and that of $v$ w.r.t. the coordinate basis. By the Smooth Manifold Chart Lemma, the tangent bundle of a smooth n-manifold admits a natural topology $\mathcal{T}_{\tilde{\phi}}$ and a natural smooth structure $\mathcal{A}_{\tilde{\phi}}$ determined by the collection of natural coordinate maps $\tilde{\phi}_\alpha: \pi^{-1}(U_\alpha) \mapsto \mathbb{R}^{2n}$ associated with each smooth chart $\phi_\alpha: U_\alpha \mapsto \mathbb{R}^n$, so that $(T M, \mathcal{T}_{\tilde{\phi}}, \mathcal{A}_{\tilde{\phi}})$ is a smooth 2n-manifold and its projection map is a smooth map. The tangent bundle of a smooth n-manifold with a global smooth chart is diffeomorphic to the product manifold $M \times \mathbb{R}^n$, a subset of $\mathbb{R}^{2n}$.

Global differential $d F$ or global tangent map $T F$ of ---or global pushforward $F_∗$ of tangent vectors by---a smooth map is the disjoint union of its pointwise differentials: $d F(p, v) = (F(p), d F_p(v))$. The global differential of a smooth map is a smooth map itself: $F \in C^\infty(M, N)$, then $d F \in C^\infty(T M, T N)$.

### Cotangent Bundle

Cotangent space $T_p^∗ M$ at a point on a smooth manifold is the dual space to the tangent space at that point: $T_p^∗ M = (T_p M)^∗$. Tangent covector at a point is an element of the cotangent space at the point.

Cotangent bundle (余切丛) $T^∗ M$ of a smooth manifold is the disjoint union of its cotangent spaces: $T^∗ M = \sqcup_{p \in M} T_p^∗ M$. The natural projection map $\pi: T^∗ M \mapsto M$ of a cotangent bundle sends each indexed tangent covector to the point of tangent: $\pi(p, \omega) = p$. Coordinate covector field $\lambda^i$ or $d x^i$ associated with a smooth chart are the maps that take each point in the domain of the chart to the covectors dual to the i-th coordinate vector associated with the chart: $\langle \lambda^i, \frac{\partial}{\partial x^j} \rangle = \delta^i_j$. Cotangent bundle $(T^∗ M, \pi, \{\Phi_\alpha\}_{\alpha \in A}))$ of a smooth manifold is the smooth vector bundle of rank $n$ over $M$, following the vector bundle chart lemma, such that the total space $T^∗ M$ is endowed with the unique topology and smooth structure for which all coordinate covector fields are smooth local sections, and the maps sending each indexed covector to its coordinate representation in the coordinate dual basis are diffeomorphisms: $\Phi_\alpha (\xi_i \lambda^i|_p) = (p, (\xi_i)_{i=1}^n)$. Natural coordinates $(x^i, \xi_i)$ of an element $(p, \omega)$ of a cotangent bundle associated with a smooth chart is the tuple of the local coordinate representation of $p$ w.r.t. the chart and that of $\omega$ w.r.t. the coordinate dual basis. Natural coordinates are smooth charts of a cotangent bundle.

Pointwise cotangent map $d F_p^∗$ of ---or pullback (拉回) $F_p^∗$ of tangent covectors by--- a smooth map $F \in C^\infty(M, N)$ at a point $p \in M$ is the dual operator $(d F_p)^∗ \in \mathcal{L}(T_{F(p)}^∗ N, T_p^∗ M)$ of its pointwise differential at that point: $\forall \omega \in T_{F(p)}^∗ N$, $\forall v \in T_p M$, $d F_p^∗ (\omega)(v) = \omega(d F_p (v))$. Global cotangent map $d F^∗$ of ---or global pullback of tangent covectors $F^∗$ by---a diffeomorphism is the map whose restriction to each cotangent space $T_q^∗ N$ equals $d F_{F^{-1}(q)}^∗$. The global pullback of a diffeomorphism is a smooth bundle homomorphism from the cotangent bundle $T^∗ N$ of its codomain to the cotangent bundle $T^∗ M$ of its domain.

### Rank

Rank of a smooth map $F$ at a point $p$, or rank of the differential $d F_p$, is the rank of the Jacobian matrix of $F$ in any smooth chart containing $p$: $\text{rank}~d F_p = \dim \text{im}(d F_p)$. A smooth map has constant rank if its rank at every point is the same, which is denoted as $\text{rank}~F$.

A smooth map has full rank if its rank at every point equals the lower of the dimensions of its domain and its codomain: $\text{rank}~F = \min\{\dim M, \dim N\}$. Smooth submersion (浸没) is a smooth map whose differentials are surjective everywhere: $\text{rank}~F = \dim N$. Smooth immersion (浸入) is a smooth map whose differentials are injective everywhere: $\text{rank}~F = \dim M$. Local diffeomorphism $F: M \mapsto N$ is a smooth map that equals gluing together some diffeomorphisms whose ranges are open subsets. A map between smooth manifolds without boundary is a local diffeomorphism if and only if it is both a smooth submersion and a smooth immersion. Rank Theorem (Canonical form of constant-rank maps): A smooth map $F$ of constant rank $r$ between smooth manifolds $M$ and $N$ of dimensions $m$ and $n$ has a local coordinate representation of the canonical form $\hat{F}(x^i)_{i=1}^m = ((x^i)_{i=1}^r, 0_{n-r})$ at each point $p \in M$ w.r.t. some smooth coordinate charts $\phi$ and $\psi$ centered at $p$ and $F(p)$. Corollary: A smooth map $F$ on a connected manifold $M$ has local coordinate representations $\hat{F}$ which are linear maps and whose domains cover $M$, if and only if $F$ has constant rank.

Global Rank Theorem: For a smooth map $F: M \mapsto N$ of constant rank: if it is surjective, then it is a smooth submersion; if it is injective, then it is a smooth immersion; if it is bijective, then it is a diffeomorphism.

Smooth embedding $F: M \mapsto N$ of a smooth manifold $M$ into another smooth manifold $N$ is a smooth immersion---not just a smooth map---that is also a topological embedding. A smooth immersion may not be a topological embedding, e.g. the curve $\beta: (-\pi, \pi) \mapsto \mathbb{R}^2$ defined by $\beta(t) = (\sin 2t, \sin t)$ maps an open interval to a lemniscate (8-shape). A smooth map is a smooth immersion if and only if it equals some smooth embeddings glued together. Although any smooth manifold admits a smooth embedding into some Euclidean space (see the strong Whitney embedding theorem), working with manifolds as abstract topological spaces can be an advantage, both in complexity and interpretability, e.g. spacetime in general relativity.

... A smooth map $\pi: M \mapsto N$ is a smooth submersion if and only if the images of smooth local sections of $\pi$ cover $M$. Any smooth submersion is an open map; if it is surjective, it is a quotient map. A map $F: N \mapsto P$ between smooth manifolds is smooth if and only if the composition of any surjective smooth submersion $\pi: M \mapsto N$ onto $N$ is smooth. If a smooth map $F: M \mapsto P$ is constant on the level sets of a surjective smooth submersion $\pi: M \mapsto N$, then there exists a unique smooth map $\tilde{F}: N \mapsto P$ such that $F = \tilde{F} \circ \pi$.

Smooth covering map $\pi: E \mapsto M$ is a covering map that is also a local diffeomorphism. Every smooth covering map is a smooth submersion, an open map, and a quotient map.

### Submanifold

Smooth submanifold $(S, \mathcal{T}, \mathcal{A})$ of a smooth manifold $M$ is a subset of the manifold endowed with a topology and a smooth structure such that it is a smooth manifold. In the context of a submanifold of a smooth manifold, we call the manifold the ambient manifold of the submanifold. Although a submanifold is a manifold itself and thus can be treated in an abstract way, we often consider it as a subset of the ambient manifold, which provides a global representation of the submanifold however the ambient manifold is locally represented in a Euclidean space.

Immersed submanifold of a smooth manifold is a smooth submanifold without boundary such that the inclusion map $\iota: S \mapsto M$ is a smooth immersion. Note that the topology of an immersed submanifold is not necessarily the subspace topology. By convention, smooth submanifold refers to immersed submanifold. Embedded submanifold of a smooth manifold is a smooth submanifold without boundary such that the topology is the subspace topology and the inclusion map is a smooth embedding. Immersed submanifolds are locally embedded. Properly embedded submanifold of a smooth manifold is an embedded submanifold such that the inclusion map is a proper map. An embedded submanifold is properly embedded if and only if it is a closed subset of the ambient manifold. Every compact embedded submanifold is properly embedded.

Codimension (余维度) $\text{codim} S$ of an embedded submanifold of a smooth manifold is the difference between their dimensions: $\text{codim} S = \dim M - \dim S$. Open submanifold $(U, \mathcal{T}_U, \mathcal{A}_U)$ of a smooth manifold $M$ is an open subset of $M$ endowed with the subspace topology and the subspace smooth structure. The open submanifolds of a smooth manifold are all its embedded submanifolds of codimension zero. Regular domain in a smooth manifold $M$ is a properly embedded codimension-0 submanifold with boundary. Embedded hypersurface is an embedded submanifold of codimension one.

The image $F(M_1)$ of any smooth embedding $F: M_1 \mapsto M_2$, endowed with the subspace topology of $\mathcal{T}_2$ and the smooth structure $\{\phi \circ F^{-1} : \phi \in \mathcal{A}_1\}$, is an embedded submanifold of $M_2$ diffeomorphic to $M_1$ via $F$; the said smooth structure is the only smooth structure such that $F$ is a diffeomorphism onto its image $F(M_1)$.

The graph $\Gamma(f)$ of any smooth map $f: U \times N$ from any open subset $U$ of any smooth n-manifold $M$ without boundary, endowed with the subspace topology and the smooth structure determined by the smooth atlas consisting of the graph coordinate chart, is a properly embedded n-submanifold of $M \times N$ diffeomorphic to the domain $U$ via $f$.

The most useful kinds of embedded submanifolds are those identified as level sets. Constant-Rank Level Set Theorem: Any level set of a smooth map of constant rank $r$ is a properly embedded submanifold in the domain with codimension $r$. (For a local, $C^k$ version, see [@Rudolph2013, Prop 1.7.6].) Submersion Level Set Theorem: Any level set of a smooth submersion is a properly embedded submanifold in the domain with codimension $\dim N$. Embedded submanifolds are locally level sets of a smooth submersion: A subset of a smooth manifold is an embedded submanifold with codimension $n$ if and only if it has an open cover consisting of level sets of smooth submersions to $\mathbb{R}^n$. Regular point and critical point $p$ of a smooth map $\Phi: M \mapsto N$ is a point in $M$ where the differential $d \Phi_p$ is surjective / not surjective. Regular value and critical value $c$ of a smooth map is a point in $N$ such that the level set $\Phi^{-1}(c)$ has no / some critical points. Regular level set is a level set consisting of regular points. Regular Level Set Theorem: Any regular level set of a smooth map is a properly embedded submanifold in $M$ with codimension $\dim N$. In other words, a level set of a smooth map $\Phi: M \mapsto N$ is a properly embedded (m-n)-submanifold in $M$ if $\Phi$ is surjective, i.e. has rank $n$, on the level set. Regular Level Set Theorem ($C^k$ version) [@Hirsch1976, Thm 3.2]: Any regular level set of a $C^k$ map between $C^k$ manifolds, $k \ge 1$, is a properly embedded $C^k$ submanifold in the domain with codimension $\dim N$. Defining map for an embedded submanifold $S$ of a smooth manifold $M$ is a smooth map $\Phi$ on $M$ such that $S$ is a regular level set. Defining function is a defining map whose codomain is a Euclidean space.

Smooth local parameterization $X: \phi(U) \mapsto M$ of a submanifold of a smooth manifold is the inverse of a smooth chart of the submanifold followed by the inclusion map: $X = \iota \circ \phi^{-1}$, where $\iota: S \mapsto M$, $\phi: U \mapsto \mathbb{R}^n$. The domain of a smooth local parameterization of an n-submanifold is an open subset of the Euclidean n-space, its closed upper half-space, or its nonnegative space, if the submanifold is without boundary, with boundary, or with corners, respectively. Smooth local parameterizations of an n-submanifold of the Euclidean m-space is a function from a subset of the Euclidean n-space into the Euclidean m-space. Graph parameterization of the graph of a smooth real-valued function $f: U \mapsto \mathbb{R}$ on an open subset of the Euclidean n-space is the smooth global parameterization $\Phi_f: U \mapsto \mathbb{R}^{n+1}$, $\Phi_f(x) = (x, f(x))$.

### Whitney Embedding and Approximation Theorems

Measure zero subset $A$ in a smooth n-manifold $M$ is a subset whose image has zero n-volume in every smooth chart: $\forall (U, \phi) \in \mathcal{A}$, $\lambda(\phi(A \cap U)) = 0$. Sard’s theorem: The set of critical values of a smooth map $F: M \mapsto N$ has measure zero in $N$.

Whitney embedding theorem [@Whitney1936]: Any smooth n-manifold admits a proper smooth embedding into $\mathbb{R}^{2n+1}$. Note that 2n comes from the dimensions of the product manifold $M \times M$ and the tangent bundle $T M$. The Whitney embedding theorem imply that the intrinsic/metric definition of smooth manifolds is no more general than the extrinsic/submanifold definition. Strong Whitney embedding theorem [@Whitney1944a]: Any smooth n-manifold, $n > 0$, admits a smooth embedding into $\mathbb{R}^{2n}$. This is the best possible embedding dimension for smooth 1- and 2-manifolds, e.g. Klein bottle is a smooth 2-manifold that always self-intersects in $\mathbb{R}^3$. However, every smooth 3-manifold can be embedded in $\mathbb{R}^5$ [@Wall1965].

Whitney immersion theorem: Any smooth n-manifold admits a smooth immersion into $\mathbb{R}^{2n}$. Strong Whitney immersion theorem: [@Whitney1944b] Any smooth n-manifold, $n > 1$, admits a smooth immersion into $\mathbb{R}^{2n-1}$. The best possible immersion dimension is provd by [@Cohen1985]: Every compact smooth n-manifold can be immersed in $\mathbb{R}^{2n-a(n)}$, where $a(n)$ is the number of 1’s in the binary expression for $n$. For example, every 3-manifold can be immersed in $\mathbb{R}^4$, every 4-manifold can be immersed in $\mathbb{R}^7$.

Normal space $N_p M$ at a point $p$ on an embedded m-submanifold $M$ of $\mathbb{R}^n$ is the (n-m)-dimensional subspace of the tangent space $T_p \mathbb{R}^n$ that consists of all vectors orthogonal to the tangent space $T_p M$ with respect to the Euclidean inner product. Normal bundle (法丛) $N M$ of an embedded submanifold $M$ of $\mathbb{R}^n$ is the disjoint union of the normal spaces at all points of $M$: $N M = \sqcup_{p \in M} N_p M$. The normal bundle of any embedded submanifold of $\mathbb{R}^n$ is an embedded n-submanifold of the tangent bundle $T \mathbb{R}^n$. Normal exponential map $E: \mathscr{E}_M \mapsto \tilde M$ of an embedded submanifold in a Riemannian manifold is the restriction of the exponential map of the ambient manifold to (the intersection of its domain with) the normal bundle of the submanifold: $E = \exp|_{\mathscr{E}_M}$, where $\mathscr{E}_M = \mathscr{E} \cap N M$. In case the ambient manifold is Euclidean, the normal exponential map can be identified with the addition map: $E(p, v) = p + v$. Normal neighborhood $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$ is a neighborhood that is diffeomorphic, via the addition map, to a fiber bundle of star-shaped neighborhoods of the origin in the normal spaces: $U = E(V)$, $E(p, v) = p + v$, $V = \{(p, v) \in NM: v \in S_p\}$, where $S_p$ is a star-shaped neighborhood of the origin in $N_p M$. Tubular neighborhood $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$ is a neighborhood that is diffeomorphic, via the addition map, to a fiber bundle of normal disks of continuous radius: $U = E(V)$, $E(p, v) = p + v$, $V = \{(p, v) \in NM: |v| < \delta(p)\}$, $\delta \in C^0(M, \mathbb{R}_+)$. ε-tubular neighborhood or uniform tubular neighborhood of radius ε is a tubular neighborhood with constant radius: $\delta(p) = \varepsilon$. Any embedded submanifold of $\mathbb{R}^n$ has a tubular neighborhood. Retraction $r: X \mapsto M$ of a topological space $X$ onto a subspace $M$ is a surjective continuous map whose restriction to the codomain is the identity map: $r|_M = \text{Id}_M$. Any tubular neighborhood $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$ has a retraction $r: U \mapsto M$ that is also a smooth submersion. Fermi coordinates $\phi: U \mapsto \mathbb{R^n}$ on a normal neighborhood of an embedded d-submanifold of a Riemannian n-manifold is a coordinate map that concatenates a local coordinate chart of the submanifold and a local orthonormal frame for the normal bundle [@Gray1982; @Gray2004]: $\phi(E(p, v)) = (x^i, v^j)_{i \in d, j \in n-d}$.

Whitney Approximation Theorem for functions [@Whitney1936]: Any continuous function on a smooth manifold can be uniformly approximated by a smooth function: $F \in C^0(M, \mathbb{R}^k)$, $F|_A \in C^\infty, A \in \mathcal{T}^∗$, $\forall \delta \in C^0(M, \mathbb{R}_+)$, $\exists \tilde{F} \in C^\infty(M, \mathbb{R}^k)$, $|\tilde{F} - F| < \delta$, $\tilde{F}|_A = F|_A$. Whitney Approximation Theorem [@Whitney1936]: Any continuous map between smooth manifolds is homotopic to a smooth map; if the map is smooth on a closed subset of the domain and the codomain has no boundary, then the homotopy can be taken to be relative to the subset: $\forall F \in C^0(M, N)$, ($F|_A \in C^\infty, A \in \mathcal{T}^∗$, $\partial N = \emptyset$), $\exists \tilde{F} \in C^\infty(M, N)$, $\tilde{F} \simeq F$, ($\tilde{F}|_A = F|_A$). Extension Lemma for Smooth Maps: A smooth map $f: A \mapsto N$ from a closed subset $A$ of a smooth manifold $M$ to a smooth manifold without boundary can be extended to a smooth function $\tilde{f}$ on $M$ if and only if it has a continuous extension on $M$. If two smooth maps are homotopic, then they are smoothly homotopic; if they are homotopic relative to a closed subset of the domain and the codomain has no boundary, then they are smoothly homotopic relative to the subset.

### Orientation

Consistently oriented bases of a finite-dimensional vector space... Orientation for a finite-dimensional vector space. Oriented vector space... A vector space has exactly two choices of orientation. Pointwise orientation on a smooth manifold... Oriented frame... Orientation on a smooth manifold is a continuous point-wise orientation, i.e. ... Orientable manifold... Nonorientable manifold. A orientable connected smooth manifold has exactly two orientations. Two orientations of a connected, orientable, smooth manifold are equal if they agree at one point. Oriented manifold... Orientation form... Oriented form... Oriented smooth chart... Consistently oriented smooth altas...

## Fiber Bundle

Fibration (aka fiber space, fiber bundle) is a way to decompose a high-dimensional manifold into a manifold-indexed collection of homeomorphic low-dimensional manifolds, endowed with a manifold topology and a smooth structure such that it is locally homeomorphic to (subpaces of) product spaces.

Commutative diagram of fiber bundle.

Fiber bundle $(X, \pi, \{\Phi_\alpha\}_{\alpha \in A})$ over a topological space $B$ is a topological space $X$ endowed with a quotient map $\pi: X \mapsto B$ and a collection of homeomorphisms $\Phi_\alpha: \pi^{-1}(U_\alpha) \mapsto U_\alpha \times F$ from saturated subsets to product spaces, whose domains form an open cover and whose first component maps match the quotient map: $\cup_\alpha U_\alpha = B$, $\cup_\alpha \Phi^1_\alpha = \pi$. We call $X$ the total space of the bundle, $B$ its base space, $\pi$ its canonical projection, topological space $F$ its model fiber, and $\Phi$ a local trivialization of $X$ over $U$. Denote $X_p$ the fiber over a point $p$ in the base space: $X_p = \pi^{-1}(p)$. Global trivialization of a fiber bundle is a local trivialization over its base space. Product fiber bundle $(B \times F, \pi_1)$ is a fiber bundle consisting of the product space of the base space and its model fiber, and the first canonical projection. Trivial fiber bundle is a fiber bundle that admits a global trivialization. Equivalently, a trivial fiber bundle is one that is homeomorphic to the product fiber bundle of its base space and its model fiber: $X \cong B \times F$. Smooth fiber bundle is a fiber bundle where $X, B, F$ are smooth manifolds, $\pi$ is a smooth map, and the local trivializations can be chosen to be diffeomorphisms. Smooth local trivialization on a smooth fiber bundle is a local trivialization that is a diffeomorphism onto its image. Smoothly trivial fiber bundle is a smooth fiber bundle that admits a smooth global trivialization. Equivalently, a smoothly trivial fiber bundle is one that is diffeomorphic to the product fiber bundle $B \times F$.

Real vector bundle $(E, \pi, \{\Phi_\alpha\}_{\alpha \in A})$ of rank $k$ over $B$ is a fiber bundle with model fiber $\mathbb{R}^k$, such that every restriction $\Phi_\alpha |_{E_p}$ of a local trivialization to a fiber is a vector space isomorphism to $\{p\} \times \mathbb{R}^k$. Real line bundle is a rank-1 vector bundle. Smooth vector bundle is a vector bundle that is a smooth fiber bundle. The Möbius bundle is a smooth line bundle over the circle that is not trivial. A smooth vector bundle is smoothly trivial if and only if it admits a smooth global frame. Transition function $\tau: U_1 \cap U_2 \mapsto \text{GL}(k, \mathbb{R})$ between smooth local trivializations on a smooth rank-$k$ vector bundle is a smooth map to invertible matrices, such that the composite map $\Phi_2 \circ \Phi_1^{-1}$, which is a transformation on $(U_1 \cap U_2) \times \mathbb{R}^k$, can be written as $\Phi_2 \circ \Phi_1^{-1} (p, v) = (p, \tau(p) v)$.

The tangent bundle of a smooth n-manifold together with its natural projection map, $(T M, \pi)$, is a smooth vector bundle of rank $n$ over the manifold. For a tangent bundle, each transition function between local trivializations associated with two smooth charts equals the Jacobian matrix of the transition map between these charts.

### Derived Fiber Bundles

Vector Bundle Construction Theorem (construction by surgery): Given an open cover $U_\alpha$ of a smooth manifold $M$ and a collection of smooth maps $\tau_{\alpha\beta}: U_\alpha \cap U_\beta \mapsto \text{GL}(k, \mathbb{R})$ that are transitive $\tau_{\alpha\beta} \tau_{\beta\gamma} = \tau_{\alpha\gamma}$, one can construct a smooth rank-$k$ vector bundle $((\sqcup_p E_p, \mathcal{T}_\tau, \mathcal{A}_\tau), \pi, \{\Phi_\alpha\}_{\alpha \in A})$ over $M$ such that the transition functions among its smooth local trivializations are $\tau_{\alpha\beta}$.

Restriction $X|_S$ of a fiber bundle to a subset $S$ of its base space is the fiber bundle $(\pi^{-1}(S), \pi|_S, \{\Phi_\alpha|_S\}_{\alpha \in A})$, where $\pi^{-1}(S)$ is a subspace of $X$. The restriction $E|_S$ of a smooth vector bundle to a smooth submanifold of its base manifold, together with its subspace smooth structure, is a smooth vector bundle. Ambient tangent bundle $T M|_S$ of a smooth submanifold $S$ over its ambient manifold $M$ is the restriction of the tangent bundle of the ambient manifold to the submanifold.

Subbundle $(D, \pi|_D, \{\Phi_\alpha|_D\}_{\alpha \in A})$ of a vector bundle $E$ is a vector bundle where $D$ is a topological subspace of $E$ such that $D_p$ is a linear subspace of $E_p$.

Whitney sum $E' \oplus E''$ of smooth rank-$k'$ and rank-$k''$ vector bundles over a smooth manifold $M$ is the rank-$(k'+k'')$ vector bundle over $M$ determined by transition functions $\tau = \tau' \oplus \tau''$ (where $\oplus$ is the direct sum of matrices): $((\sqcup_p (E'_p \oplus E''_p), \mathcal{T}_\tau, \mathcal{A}_\tau), \pi)$, where $\oplus$ is the direct sum of vector spaces.

### Section

Global section or cross section $\sigma$ of a vector bundle is a section of its canonical projection: $\sigma \in C(B, E)$, $\pi \circ \sigma = \text{Id}_B$. Local section of a vector bundle is a local section of its canonical projection: $\sigma \in C(U, E)$, $U \subset B$, $\pi \circ \sigma = \text{Id}_U$. Smooth section of a vector bundle is a section that is a smooth map. The set $\Gamma(E)$ of all smooth (global) sections of a smooth vector bundle is a vector space under pointwise addition and scalar multiplication. Rough section of a vector bundle is a right inverse of the canonical projection over a subset of the base space: $\sigma: U \mapsto E$, $U \subset B$, $\pi \circ \sigma = \text{Id}_U$. Zero section $\zeta$ of a vector bundle is the global section that takes every point in the base space to the zero vector at that point: $\forall p \in B$, $\zeta(p) = 0 \in E_p$. Support of a section is the closure of its non-zero set: $\overline{B \setminus \sigma^{-1}(0)}$.

Local frame $(E_i)_{i=1}^n$ for a smooth n-manifold is an ordered n-tuple of vector fields on an open subset of the manifold, which is linearly independent and spans the vector bundle, i.e. it is a basis for the fiber space at each point. Global frame is a frame defined on the entire manifold. Smooth frame is a frame consisting of smooth vector fields.

### Bundle Homomorphism

Fiber-space morphism from a fiber space $\pi_1: X_1 \mapsto B_1$ into another $\pi_2: X_2 \mapsto B_2$ is a map $\phi: X_1 \mapsto X_2$ such that $\pi_2 \circ F \circ \pi_1^{-1}$ is a map $f: B_1 \mapsto B_2$. Fiber-space isomorphism is a morphism between $\pi_1$ and $\pi_2$ that is a homeomorphism (topological isomorphism) between $X_1$ and $X_2$.

Bundle homomorphism (丛同态) $F: E \mapsto E'$ between vector bundles is a continuous map that is linear on each fiber and there is a map $f: B \mapsto B'$ between the base spaces such that $\pi' \circ F = f \circ \pi$. The bundle homomorphism $F$ is said to cover the map $f$ between the base spaces. The map covered by a bundle homomorphism is continuous and unique. The map covered by a smooth bundle homomorphism between smooth vector bundles is smooth. Bundle isomorphism (丛同构) is a bijective bundle homomorphism whose inverse is also a bundle homomorphism. Smooth bundle isomorphism is a bundle isomorphism that is a diffeomorphism.

Bundle homomorphism between vector bundles over the same base space is bundle homomorphism that covers the identity map of the base space. Any bijective smooth bundle homomorphism over a smooth manifold is a smooth bundle isomorphism.

## Algebraic Topology

Algebraic topology is the subject that studies topological properties of topological spaces by attaching algebraic structures to them in a topologically invariant way, see Abstract Algebra.

### Complex

Complex (复形). Cell complex (胞腔复形). CW complex. Chain complex (链复形). Cochain complex (上链复形).

Simplicial complex (单纯复形) $K$ is a space with a triangulation: $K = \{s_i\}_i \subset \mathbb{R}^n$ is a class of simplices such that every face of a simplex $s_i$ is in $K$, and the intersection of any two simplices is a face of each of them. Graph (in discrete math) as a topological space is equivalent to simplicial 1-complex.

### Lie Group

Topological group $(G, ∗, \mathcal{T})$ is a group with a topology such that the group operation and the inverse map are continuous: $∗ \in C(G^2, G)$, $∗^{-1}(e) \in C(G, G)$. Lie group $(G, ∗, \mathcal{T}, \mathcal{A})$ is a smooth manifold with a group structure such that the multiplication and inversion maps are smooth maps. Matrix Lie group is a subgroup of the complex general linear group $\text{GL}_n(\mathbb{C})$, e.g. the real general linear group $\text{GL}_n(\mathbb{R})$ and the orthogonal group O(n). Every element in a matrix Lie group, as a left action, is a diffeomorphism: $\forall A \in G$, let $L_A(B) = A B$, then $L_A: G \cong G$. Representation $\rho: G \mapsto \text{GL}(V)$ of a finite-dimensional Lie group is a Lie group homomorphism to the general linear group on a finite-dimensional vector space. Faithful representation is an injective representation. Exponential map $\exp: \mathfrak{g} \mapsto G$.

Lie bracket $[X, Y]$ of a pair of smooth vector fields on a smooth manifold is the smooth vector field such that applying it to any smooth function equals the difference in the functions obtained by applying the pair to the function in different orders: $\forall f \in C^\infty(M)$, $[X, Y] f = X Y f - Y X f$. The Lie bracket of any pair of smooth vector fields is a smooth vector field: $\forall X, Y \in \mathfrak{X}(M)$, $[X, Y] \in \mathfrak{X}(M)$. The Lie bracket of a pair of smooth vector fields has coordinate representation: $[X, Y] = \left(X^i \frac{\partial Y^j}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i}\right) \frac{\partial}{\partial x^j}$. The Lie bracket operator on smooth vector fields is bilinear, anti-symmetric, and satisfies: (1) Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$; (2) $[f X, g Y] = f g [X, Y] + (f X g) Y - (g Y f) X$, where $f, g \in C^\infty(M)$. Lie algebra $(\mathfrak{g}, (+, \cdot_{\mathbb{R}}), [\cdot, \cdot])$ over real numbers is a real vector space endowed with a map (called the bracket) that is bilinear, anti-symmetric, and satisfies the Jacobi identity. Lie algebra $\text{Lie}(G)$ of a Lie group G is the Lie algebra of all left-invariant vector fields on the Lie group. The Lie algebra $(T_I G, (+, \cdot_{\mathbb{R}}), [\cdot, \cdot])$ of a matrix Lie group G is its tangent space at the identity endowed with the matrix commutator as the bracket: $[V, W] = V W - W V$, where $V, W \in T_I G$. Note that the tangent space of a matrix Lie group G at an element A can be written as: $T_A G = A \text{Lie}(G) = \{A V \in M_n : V \in T_I G\}$. The Lie algebra $\mathfrak{so}(n)$ of the orthogonal group consists of the set of skew-symmetric matrices: $\mathfrak{so}(n) := \text{Lie}(O(n)) = T_I O(n) = \Omega(n)$, where $\Omega(n) = \{\Omega \in M_n : \Omega = - \Omega^T \}$.

Lie derivative $\mathscr{L}_V W$ of $W$ w.r.t. $V$, both smooth vector fields on a smooth manifold, is the rough vector field on the manifold defined by: $(\mathscr{L}_V W)_p = \frac{d}{d t}\bigg|_{t=0} d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)})$, i.e. $(\mathscr{L}_V W)_p = \lim_{t \to 0} (d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)}) - W_p) / t$, where $\theta$ the flow of $V$. Lie derivative equals Lie bracket: $\forall V, W \in \mathfrak{X}(M)$, $\mathscr{L}_V W = [V, W]$.

Closed subgroup theorem: Every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. There is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups.

Left action of a topological group G on a topological space X is a map $\phi : G \times X \mapsto X$ such that: $\forall g_1, g_2 \in G$, $\forall p \in X$, $\phi(g_1, \phi(g_2, p)) = \phi(g_1 g_2, p)$ and $\phi(e, p) = p$ (e is the identity element of G). If the action is unambiguous, one may simply write: $\phi(g, p) = g \cdot p$; in comparison, the group operation is omitted altogether: $g_1 ∗ g_2 = g_1 g_2$. Note that since the map takes two variables of distinct types, for definiteness, one usually put the group element first, i.e. the group acts on the left; similarly, a right action is an action where the group acts on the right. Orbit $G \cdot p$ of a point $p \in X$ is the image of the group acting on the point: $G \cdot p = \{g \cdot p : g \in G\}$. Orbit space $X / G$ of an action is the quotient space of orbits: $X / G = \{G \cdot p : p \in X\}$; here the quotient map $\pi: X \mapsto X / G$ takes each point to its orbit $\pi(p) = G \cdot p$. Isotropy group, stationary subgroup, or stabilizer $G_p$ of a point p in a set X with action G is the subgroup where the point is a fixed point: $G_p = \{g \in G : g \cdot p = p\}$.

Smooth action of a Lie group on a smooth manifold is an action that is a smooth map. Free action is one such that every non-identity element in the group acts without fixed points: if $\exists p \in M$, g · p = p, then g = e; or equivalently, every isotropy group is trivial: $\forall p \in M$, $G_p = \{e\}$. Proper action is one such that each compact subset is moved away from itself by most elements of the group. Every continuous action by a compact Lie group on a manifold is proper. Quotient manifold theorem: If a Lie group G acts smoothly, freely, and properly on a smooth manifold M, then the orbit space M/G is a smooth manifold of dimension dim(M) - dim(G), whose smooth structure is uniquely determined such that the quotient map $\pi: M \mapsto M / G$ is a smooth submersion. In this case, we call the orbit space M/G a quotient manifold.

Equivariant map between smooth manifolds with smooth G-actions is a map that commutes with the action: $\forall g \in G$, $F(g \cdot p) = g \cdot F(p)$; order is switch for right actions. Equivariant rank theorem: Let M and N be smooth manifolds with smooth G-actions. If the G-action on M is transitive (i.e. (M, G) is a homogeneous G-space), then any smooth equivariant map $F: M \mapsto N$ constant rank. (Therefore the global rank theorem applies.)

Orbits of proper actions: Given a proper smooth action ϕ of a Lie group G on a smooth manifold M, the orbit map $\phi^{(p)}: G \mapsto M$ of any point p ∈ M is a proper map, and thus the orbit G·p is closed in M. If its isotropy group $G_p$ contains only the identity element, then the orbit map is a smooth embedding, and the orbit is a properly embedded submanifold. Theorem: If a compact Lie group G acts smoothly on a smooth manifold M, then the orbit G·p of any point p ∈ M is a properly embedded submanifold of M, and it is diffeomorphic to the quotient manifold $G / G_p = \{G_p g : g \in G\}$. Note that $G / G_p$ is a quotient manifold because the isotropy group acts smoothly and freely on the compact Lie group, where the action is the group operation.

Every Lie group admits a left-invariant smooth global frame.

Homogeneous G-space, homogeneous space, or homogeneous manifold (M, G) is a smooth manifold M with a smooth transitive action by a Lie group G: (1) $\forall x, y \in M, \exists g \in G: g \cdot x = y$ (transitivity); (2) $e \cdot x = x$ (identity map); (3) $(g h) \cdot x = g \cdot (h \cdot x)$ (composition); The elements of G are called the symmetries of M. The isotropy group $G_p$ of any point p in a homogeneous G-space is a closed subgroup of G. Left coset $g H$ of a Lie subgroup H of a Lie group G is the subset $g H = \{g h : h \in H\}$, where $g \in G$. Left coset space of G modulo H, denoted by $G / H$, is the quotient space of all left cosets of H, that is, $G / H = \{g H : g \in G\}$. We say two elements $g_1, g_2 \in G$ are congruent modulo H, $g_1 \equiv g_2 (\text{mod} H)$, if they are in the same left coset of H, or equivalently $g_1^{-1} g_2 \in H$ (here $g^{-1}$ denotes an inverse element). Homogeneous space construction theorem: Given any closed subgroup H of G, the left coset space (G / H, G) is a homogeneous space. Homogeneous space characterization theorem: For every homogeneous space (M, G), let p be an arbitrary point in M, then (M, G) can be identified with the left coset space $(G / G_p, G)$, and $F: G / G_p \mapsto M$ defined by $F(g G_p) = g \cdot p$ is an equivariant diffeomorphism.

Reductive (homogeneous) space is a homogeneous space G / H of a connected Lie group G such that there is a subspace of $\mathfrak{g}$ that is invariant under $\text{Ad}_{\mathfrak{g}}(H)$ and complementary to $\mathfrak{h}$. Here, $\mathfrak{g}$ and $\mathfrak{h}$ are the Lie algebras of G and H respectively, and $\text{Ad}_{\mathfrak{g}}$ is the adjoint representation of $\mathfrak{g}$. Every homogeneous Riemannian space is reductive.

Examples. General linear group $\text{GL}(n, \mathbb{F})$ is a Lie group under matrix multiplication. The group $\text{GL}(V)$ of vector space isomorphisms on any n-dimensional vector space over the field $\mathbb{F}$ is a Lie group isomorphic to the general linear group $\text{GL}(n, \mathbb{F})$. Special linear group $\text{SL}(n, \mathbb{F})$ is a Lie subgroup of the general linear group. Orthogonal group $O(n)$ / unitary group $U(n)$ / special orthogonal group $SO(n)$ / special unitary group $SU(n)$ is a Lie subgroup of the real / complex, general / special linear group.

Direct product group $\prod_{i=1}^k G_i$ of Lie groups is the Lie group consisting of the product manifold and the component-wise group multiplication: $(g_i)_{i=1}^k (g_i')_{i=1}^k = (g_i g_i')_{i=1}^k$. Semidirect product group $N \rtimes_\theta H$ of Lie groups determined by a smooth left action by automorphisms $\theta: H \times N \mapsto N$ is the Lie group consisting of the product manifold and the group multiplication defined by $(n, h)(n', h') = (n (h n'), h h')$.

### Fundamental Group

Homotopy (同伦) $H: X \times I \mapsto Y$, $I = [0, 1]$, between two continuous maps $f, g: X \mapsto Y$ between the same two topological spaces is a continuous map that matches the two maps on the boundaries: $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$; denoted as $H: f \simeq g$. Two continuous maps $f, g: X \mapsto Y$ are homotopic, denoted as $f \simeq g$, if there is a homotopy between them. For any pair of topological spaces $X$ and $Y$, homotopy is an equivalence relation on the set $C^0(X, Y)$ of all continuous maps between them; the set of homotopy classes of continuous maps from $X$ to $Y$ is denoted by $[X, Y]$. Homotopy inverse $\psi: Y \mapsto X$ for a continuous map $\phi: X \mapsto Y$ is a continuous map such that their compositions are homotopic to identity maps: $\psi \circ \phi \simeq \text{Id}_X$ and $\phi \circ \psi \simeq \text{Id}_Y$. Homotopy equivalence $\phi: X \mapsto Y$ is a continuous map that has a homotopy inverse. Two topological spaces $X$ and $Y$ are homotopy equivalent, denoted as $X \simeq Y$, if there is a homotopy equivalence between them. Homotopy equivalence is an equivalence relation on the class of all topological spaces. Homotopy invariant of a homotopy equivalent class of topological spaces is a property common to all these spaces. Any homeomorphism is a homotopy equivalence, and thus topological equivalence is a finer equivalence relation than homotopy equivalence, and homotopy invariants are topological invariants. Examples of homotopy invariants: Euler characteristic, fundamental groups, singular homology groups, De Rham cohomology groups, homotopy groups.

Smooth homotopy between two smooth maps between the same two smooth manifolds is a homotopy that is also a smooth map, in the sense that it extends to a smooth map on some neighborhood of $X \times I$ in $X \times \mathbb{R}$. Isotopy (同痕) $H: X \times I \mapsto Y$ between two embeddings of $X$ into $Y$ is a homotopy such that for all $t \in (0, 1)$, $H(x, t)$ is an embedding.

Homotopy relative to, or stationary on, a subset of the domain of the related maps is one such that at any time its restriction to the subset is the same: $\forall x \in A \subset X$, $\forall t \in I$, $H(x, t) = H(x, 0)$. Two continuous maps matching on a subset of their domain are homotopic relative to the subset if there is a homotopy between them relative to the subset. Given a homotopy that is not stationary on any subset, we say the two homotopic maps are freely homotopic. Path homotopy $H: I \times I \mapsto X$ between two paths $f, g: I \mapsto X$ sharing the same endpoints in a topological space is a homotopy that is stationary at the endpoints: $H(0, t) = f(0) = g(0)$ and $H(1, t) = f(1) = g(0)$; denoted as $H: f \sim g$. Two paths sharing the same endpoints are path-homotopic, denoted as $f \sim g$, if there is a path homotopy between them.

Fundamental group $\pi_1(X, p)$ of a topological space $X$ based at a point $p$ is the set of path classes of loops based at $p$. Two paths $f, g: I \mapsto X$ are composable if they match in tandem: $f(1) = g(0)$. Product $f \cdot g$ of two composable paths $f$ and $g$ is the path connecting them in tandem: $f \cdot g (s) = f(2s), s \in [0,1/2]; g(2s-1), s \in [1/2, 1]$. The fundamental group $\pi_1(X, p)$ of any topological space $X$ at any point $p \in X$, together with path product $\cdot$, is a group. The fundamental group, in a certain sense, measures the number of holes in a topological space. Homeomorphic spaces have isomorphic fundamental groups. Homotopy equivalent spaces have isomorphic fundamental groups.

Homotopy Lifting Property: The lift of a homotopy $H: Y \times I \mapsto X$ between continuous maps from a locally connected space, given a covering map $q: E \mapsto X$, is uniquely determined by its "initial values": $\tilde H_1(x, 0) = \tilde H_2(x, 0)$ then $\tilde H_1 = \tilde H_2$. If the homotopy is stationary on some subset of its domain, then so are its lifts. Path Lifting Property: The lift of a path $f: I \mapsto X$, given a covering map $q: E \mapsto X$, is uniquely determined by its initial value: $\tilde f_1(0) = \tilde f_2(0)$ then $\tilde f_1 = \tilde f_2$. Monodromy Theorem: Given two paths with the same endpoints and a covering map, their lifts with the same initial value are path-homotopic if and only if they are path-homotopic.

Two embedded submanifolds of a smooth manifold intersect transversely (横截交) if their tangent spaces span the full tangent space whereever they intersect: $S \pitchfork S'$ if $\forall p \in S \cap S'$, $\text{Span} (T_p S \cup T_p S') = T_p M$. A smooth map is transverse (横截) to an embedded submanifold of its codomain if the pushforward of the domain tangent spaces by the map and the submanifold tangent spaces span the codomain tangent spaces whereever they intersect: $F: N \mapsto M$, $S \subset M$, $F \pitchfork S$ if $\forall p \in F^{-1}(S)$, $\text{Span} (d F_p(T_p N) \cup T_{F(p)} S) = T_{F(p)} M$. Two embedded submanifolds intersect transversely if and only if the inclusion of either one is transverse to the other. (A generalization of the regular level set theorem:) If two embedded submanifolds intersect transversely, then their intersection is an embedded submanifold whose codimension is the sum of those of the two submanifolds. Transversality homotopy theorem: Given an embedded submanifold $S$ in $M$, every smooth map $f: N \mapsto M$ is homotopic to a smooth map $g: N \mapsto M$ that is transverse to $S$.

Simply connected topological space is a path-connected topological space such that the fundamental group based at some point is the trivial group consisting of only the path class of the constant path at the point. For example, a sphere is simply connected, but a torus is not. Simple connectivity is a topological invariant.

### Functor

Functor. Fundamental group is a functor.

There are a number of functorial ways of associating to each topological space an algebraic object such as a group or a vector space, so that homeomorphic spaces have isomorphic objects. Most of these measure the existence of “holes” in different dimensions in a certain sense.

Homology (同调), cohomology (上同调). Simplicial homology (by H. Poincaré). Singular homology (by O. Veblen). Spectral homology (P.S. Aleksandrov).

De Rham cohomology group. De Rham groups are homotopy invariants, and thus topological invariants and diffeomorphism invariants. Mayer–Vietoris theorem. Degree of a continuous map between connected, compact, oriented, smooth n-manifolds.

A free abelian group has finite rank if it has a finite basis; otherwise, we say it has infinite rank. Rank of a free abelian group of a finite rank is the number of elements in any finite basis.

Singular homology group $H_p(X)$, $p \in \mathbb{N}$, is the quotient group $H_p (X) = Z_p(X) / B_p(X)$, i.e. $H_p (X) = \text{Ker}\partial_p / \text{Im}\partial_{p+1}$. A singular homology group is an Abelian group which partially counts the number of holes in a topological space. Singular cohomology groups. Singular homology can be computed by restricting attention only to smooth simplices. de Rham Theorem: Integration of differential forms over smooth simplices induces isomorphisms between the de Rham groups and the singular cohomology groups.

Betti number $\beta_p(X)$ of a topological space, $p \in \mathbb{N}$, is the rank of its p-th singular homology group: $\beta_p(X) = \text{rank}~H_p(X)$. Euler characteristic $\chi(X)$ of a topological space is the alternating sum of its Betti numbers: $\chi(X) = \sum_{p \in \mathbb{N}} (-1)^p \beta_p(X)$. The Euler characteristic is a homotopy invariant.

Functorial classification of manifolds: The union of small balls around data points on the manifold $\hat{M} = \cup_i B(X_i, \varepsilon)$ has the same homology as the manifold $M$ with high probability, as long as $M$ has positive reach and $\varepsilon$ is small relative to the reach [@Niyogi2008].