Definitions

Limit point (Tag D-Y6JAS)

Let \(X\) be a topological space, \(A\subseteq X\), and \(x\in X\). The point \(x\) is a limit point of \(A\) if every neighborhood of \(x\) meets \(A\sm\ts{x}\). A limit point is also called an accumulation point.

Basis for a topology (Tag D-WKURJ)

Let \(X\) be a set. A collection \(\mcb\) of subsets of \(X\) is a basis for a topology on \(X\) if

• every \(x\in X\) lies in some \(B\in\mcb\), and

• whenever \(x \in B_1 \intersect B_2\) with \(B_1, B_2\in\mcb\), there exists \(B_3\in\mcb\) with \(x\in B_3 \subseteq B_1 \intersect B_2\).

The topology generated by \(\mcb\) is the collection of all \(U\subseteq X\) such that for each \(x\in U\) there exists \(B\in\mcb\) with \(x\in B \subseteq U\).

Boundary of a subset (Tag D-3KS2F)

Let \(X\) be a topological space and \(A\subseteq X\). The boundary of \(A\) is \(\bd A \coloneqq \cl_X(A) \sm A^\circ\), where \(\cl_X(A)\) is the closure of \(A\) and \(A^\circ\), its interior, is the set of interior points of \(A\).

Bounded set in a metric space (Tag D-2GCTV)

Let \((X, d)\) be a metric space. A subset \(S \subseteq X\) is bounded if there exists \(m\in \RR\) such that \(d(x, y) < m\) for all \(x, y\in S\).

Comparability of topologies (Tag D-KW52R)

Let \(\tau_1\) and \(\tau_2\) be topologies on a set \(X\). The topology \(\tau_1\) is finer than \(\tau_2\) if \(\tau_2\subseteq\tau_1\), and coarser than \(\tau_2\) if \(\tau_1\subseteq\tau_2\). The topologies \(\tau_1\) and \(\tau_2\) are comparable if \(\tau_1\subseteq\tau_2\) or \(\tau_2\subseteq\tau_1\).

Characterizations of connectedness (Tag D-YO6NZ)

Let \(X\) be a topological space. A separation of \(X\) is a pair \((U, V)\) of disjoint nonempty open subsets of \(X\) with \(X = U\union V\). The space \(X\) is disconnected if it has a separation, and connected otherwise.

Comparability of topologies (Tag D-MVBYO)

Let \(\tau_1\) and \(\tau_2\) be topologies on a set \(X\). The topology \(\tau_1\) is finer (also stronger or larger) than \(\tau_2\) if \(\tau_2\subseteq\tau_1\), and coarser (also weaker or smaller) than \(\tau_2\) if \(\tau_1\subseteq\tau_2\). The topologies \(\tau_1\) and \(\tau_2\) are comparable if \(\tau_1\subseteq\tau_2\) or \(\tau_2\subseteq\tau_1\).

Connected components (Tag D-FAEYE)

Let \(X\) be a topological space. For \(x, y\in X\) write \(x\sim y\) if there is a connected subspace \(C\subseteq X\) with \(x, y\in C\). The equivalence classes of \(\sim\) are the connected components of \(X\).

Closed and open maps (Tag D-Q5272)

Let \(X\) and \(Y\) be topological spaces and \(f\colon X\to Y\) a map. The map \(f\) is a closed map if \(f(A)\) is closed in \(Y\) for every closed \(A\subseteq X\), and an open map if \(f(U)\) is open in \(Y\) for every open \(U\subseteq X\) [@Mun00].

Compact space (Tag D-EILKJ)

A topological space \((X, \tau)\) is compact if every open cover of \(X\) has a finite subcover: for every family \((U_j)_{j\in J}\) in \(\tau\) with \(X = \bigcup_{j\in J} U_j\), there is a finite subset \(J'\subseteq J\) with \(X = \bigcup_{j\in J'} U_j\).

Closed set (Tag D-UI7ZL)

Let \(X\) be a topological space. A subset \(U \subseteq X\) is closed in \(X\) if its complement \(X\sm U\) is open in \(X\). If \(Y \subseteq X\) carries the subspace topology and \(U \subseteq Y\), then \(U\) is closed in the subspace \(Y\) if \(Y\sm U\) is open in \(Y\).

Closed map (Tag D-METXE)

Let \(X\) and \(Y\) be topological spaces. A map \(f\colon X\to Y\) is closed if \(f(C)\) is closed in \(Y\) for every closed subset \(C\subseteq X\).

Closure of a set (Tag D-ASXW6)

Let \(X\) be a topological space and \(U\subseteq X\). The closure of \(U\) in \(X\) is the intersection of all closed subsets of \(X\) containing \(U\), \[ \cl_X(U) \coloneqq \Intersect_{\substack{B\supseteq U \\ B \text{ closed in } X}} B ,\] also written \(\bar U\).

Compact as Hausdorff and quasicompact (Tag D-L6SGU)

A topological space \((X, \tau)\) is compact if \(X\) is Hausdorff and quasicompact, that is, every open cover of \(X\) has a finite subcover: for every family \((U_j)_{j\in J}\) in \(\tau\) with \(X = \bigcup_{j\in J} U_j\), there is a finite subset \(J'\subseteq J\) with \(X = \bigcup_{j\in J'} U_j\).

Continuous map (Tag D-AEAAD)

Let \(X\) and \(Y\) be topological spaces. A map \(f\colon X\to Y\) is continuous if \(\inverseof{f}(U)\) is open in \(X\) for every open set \(U \subseteq Y\).

Cover of a space or subset (Tag D-AOJG3)

Let \(X\) be a set and \(\mathcal U = \ts{U_\alpha}_{\alpha\in I}\) a collection of subsets of \(X\). The collection \(\mathcal U\) is a cover of \(X\) if \(X = \Union_{\alpha\in I} U_\alpha\), and a cover of a subset \(A\subseteq X\) if \(A\subseteq \Union_{\alpha\in I} U_\alpha\). If \(X\) is a topological space and every \(U_\alpha\) is open, \(\mathcal U\) is an open cover.

Dense subspace (Tag D-KJBAK)

Let \(X\) be a topological space. A subset \(Q\subseteq X\) is dense in \(X\) if every neighborhood of every point \(x\in X\) intersects \(Q\).

Diameter of a subset of a metric space (Tag D-HS6DE)

Let \((X, d)\) be a metric space and \(A\subseteq X\) a nonempty subset. The diameter of \(A\) is \(\diam(A)\coloneqq\sup_{p, q\in A} d(p, q)\in[0,\infty]\).

Discrete topology (Tag FD-AHIOS)

Let \(X\) be a set. The discrete topology on \(X\) is the topology \(\tau = \mathcal{P}(X)\), in which every subset of \(X\) is open.

Indiscrete topology (Tag FD-EQPN4)

Let \(X\) be a set. The indiscrete topology on \(X\) is the topology \(\tau = \ts{\emptyset, X}\).

First countable space (Tag D-E2NWN)

A topological space \(X\) is first countable if every point \(x\in X\) has a countable neighborhood basis.

Hausdorff space (Tag D-ZFRV4)

A topological space \(X\) is Hausdorff, or \(T_2\), if for all \(p, q \in X\) with \(p\neq q\) there exist disjoint open sets \(U, V\subseteq X\) with \(p\in U\) and \(q\in V\).

Injective map (Tag D-BUAYX)

Let \(A\) and \(B\) be sets. A map \(\iota\colon A\to B\) is injective if for all \(x, y\in A\), \(\iota(x) = \iota(y)\) implies \(x = y\).

Injective map (Tag D-3ZBXG)

Let \(A\) and \(B\) be sets. A map \(\iota\colon A\to B\) is injective if for all \(x, y\in A\), \(\iota(x) = \iota(y)\) implies \(x = y\).

Interior point and interior (Tag D-5VCMJ)

Let \(X\) be a topological space and \(A\subseteq X\). A point \(p\in A\) is an interior point of \(A\) if there exists a neighborhood \(U\) of \(p\) with \(U\subseteq A\). The interior \(A^\circ\) of \(A\) is the set of interior points of \(A\).

Isolated point (Tag D-D7AFV)

Let \(X\) be a topological space and \(A\subseteq X\). A point \(p\in A\) is an isolated point of \(A\) if \(p\) is not a limit point of \(A\).

Limit point compact space (Tag FD-PMA24)

A topological space \(X\) is limit point compact if every infinite subset of \(X\) has a limit point in \(X\).

Locally connected space (Tag D-GYBZ2)

A topological space \(X\) is locally connected if for every \(x\in X\) and every neighborhood \(U\) of \(x\) there is a connected open set \(V\) with \(x\in V\subseteq U\).

Locally compact space (Tag D-5EOQZ)

A topological space \(X\) is locally compact if every \(x\in X\) has a neighborhood \(U\) contained in a compact subset \(K\subseteq X\).

Locally finite collection of subsets (Tag D-6JJJU)

Let \(X\) be a topological space. A collection \(\mcs\) of subsets of \(X\) is locally finite if every \(x\in X\) has a neighborhood \(U\) such that \(U\intersect S\neq\emptyset\) for only finitely many \(S\in\mcs\).

Local homeomorphism (Tag FD-ONITX)

Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a local homeomorphism if every \(x\in X\) has an open neighborhood \(U_x\) such that \(f(U_x)\) is open in \(Y\) and \(f\vert_{U_x}\colon U_x \to f(U_x)\) is a homeomorphism.

Locally homeomorphic spaces (Tag FD-2XUJ5)

Let \(X\) and \(Y\) be topological spaces. The space \(X\) is locally homeomorphic to \(Y\) if every \(x\in X\) has an open neighborhood \(U_x\) that is homeomorphic to an open subset of \(Y\).

Locally path-connected space (Tag D-7JSLO)

A topological space \(X\) is locally path-connected if for every \(x\in X\) and every neighborhood \(U\) of \(x\) there exists a path-connected open set \(V\) with \(x\in V\subseteq U\).

Neighborhood (Tag D-JMRPA)

Let \(X\) be a topological space and \(x\in X\). A neighborhood of \(x\) is an open set \(U\subseteq X\) with \(x\in U\).

Normal space (Tag D-YEQC3)

A topological space \(X\) is normal if for all disjoint closed subsets \(A, B\subseteq X\) there exist disjoint open sets \(U, V\subseteq X\) with \(A\subseteq U\) and \(B\subseteq V\).

Neighborhood basis (Tag D-EMJTU)

Let \(X\) be a topological space and \(p\in X\). A neighborhood basis at \(p\) is a collection \(\mathcal B_p\) of neighborhoods of \(p\) such that every neighborhood \(N\) of \(p\) contains some \(B\in\mathcal B_p\).

Open and closed maps (Tag D-CTGON)

Let \(X\) and \(Y\) be topological spaces and \(f\colon X\to Y\) a function. Then \(f\) is an open map if \(f(U)\) is open in \(Y\) for every open \(U\subseteq X\), and \(f\) is a closed map if \(f(C)\) is closed in \(Y\) for every closed \(C\subseteq X\).

Paracompact space (Tag D-HQSEQ)

A topological space \(X\) is paracompact if every open cover of \(X\) has a locally finite open refinement that covers \(X\).

Quotient map (Tag D-TNBFZ)

Let \(X\) and \(Y\) be topological spaces. A map \(q\colon X\to Y\) is a quotient map if

1. \(q\) is surjective, and

2. for every \(U\subseteq Y\), \(U\) is open in \(Y\) if and only if \(\inverseof{q}(U)\) is open in \(X\)

[@Mun00].

Quotient map (Tag D-ITBUT)

Let \(X\) and \(Y\) be topological spaces. A map \(q\colon X\to Y\) is a quotient map if \(q\) is surjective and, for every \(U\subseteq Y\), \(U\) is open in \(Y\) if and only if \(\inverseof{q}(U)\) is open in \(X\).

Path-connected space (Tag D-X73EB)

Let \(X\) be a topological space and \(I = [0,1]\). The space \(X\) is path connected if for every pair of points \(x\neq y\) in \(X\) there exists a path \(f\colon I \to X\) with \(f(0) = x\) and \(f(1) = y\).

Path components (Tag D-VZFJQ)

Let \(X\) be a topological space. For \(x, y\in X\), put \(x\sim y\) if there exists a path-connected subspace \(U\subseteq X\) with \(x, y\in U\). The equivalence classes of \(\sim\) are the path components of \(X\).

Precompact subset (Tag D-7ALR2)

Let \(X\) be a topological space. A subset \(A\subseteq X\) is precompact if its closure \(\cl_{X}(A)\) is compact.

Product topology on $X\times Y$ (Tag D-JKH35)

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. The product topology on \(X\times Y\) is the topology generated by the basis \[ \mathcal B\coloneqq\ts{U\times V \st U\in\tau_X,\ V\in\tau_Y}: \] a subset \(W\subseteq X\times Y\) is open if and only if it is a union of elements of \(\mathcal B\).

Proper map (Tag D-GDXFZ)

Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is proper if \(\inverseof{f}(K)\) is compact for every compact \(K\subseteq Y\).

Quasicompact (Tag D-MQSFD)

A topological space \(X\), not necessarily Hausdorff, is quasicompact if every open cover of \(X\) has a finite subcover. A quasicompact space is compact if it is also Hausdorff.

Product and box topologies (Tag D-C5THN)

Let \((X_i)_{i\in I}\) be a family of topological spaces, let \(X \coloneqq \prod_{i\in I} X_i\), and for \(j\in I\) let \(p_j\colon X\to X_j\) be the projection. The box topology on \(X\) is the topology generated by the basis \[ \mathcal B_{\Box} \coloneqq \ts{ \prod_{i\in I} U_i \st U_i \text{ open in } X_i \text{ for every } i }. \] The product topology on \(X\) is the topology generated by the basis \[ \mathcal B_{\prod} \coloneqq \ts{ \prod_{i\in I} U_i \st U_i \text{ open in } X_i \text{ for every } i, \text{ and } U_i = X_i \text{ for all but finitely many } i }. \]

Refinement of a cover (Tag D-P6XCN)

Let \(X\) be a topological space and let \(\mathcal U\) and \(\mathcal V\) be covers of \(X\). The cover \(\mathcal V\) is a refinement of \(\mathcal U\) if for every \(V\in\mathcal V\) there exists \(U\in\mathcal U\) with \(V\subseteq U\).

Regular space (Tag D-EPTMG)

A topological space \(X\) is regular if for every closed set \(F\subseteq X\) and every point \(x\in X\sm F\) there are disjoint open sets \(U, V\subseteq X\) with \(x\in U\) and \(F\subseteq V\).

Retract (Tag D-NCLVD)

Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A retraction of \(X\) onto \(A\) is a continuous map \(r\colon X\to A\) with \(r\circ\iota=\id_A\), that is, \(r(a)=a\) for every \(a\in A\):

The subspace \(A\) is a retract of \(X\) if a retraction of \(X\) onto \(A\) exists.

Saturated subset (Tag D-KWWVL)

Let \(p\colon X\to Y\) be a surjective map. A subset \(U\subseteq X\) is saturated with respect to \(p\) if for every \(y\in Y\) with \(U\cap \inverseof{p}(y)\neq\emptyset\) we have \(\inverseof{p}(y)\subseteq U\).

Separable space (Tag D-3O6QH)

A topological space \(X\) is separable if there exists a countable subset \(Q\subseteq X\) that is dense in \(X\).

Sequentially compact space (Tag FD-DXTBN)

A topological space \(X\) is sequentially compact if every sequence in \(X\) has a subsequence that converges to a point of \(X\).

Second-countable space (Tag D-23NTI)

A topological space \(X\) is second-countable if there exists a countable basis for its topology.

Subspace topology (Tag D-LB2LS)

Let \((X, \tau)\) be a topological space and \(A\subseteq X\) a subset. The subspace topology on \(A\) is \[ \tau_A\coloneqq\ts{A\cap U \st U\in\tau}. \]

Surjection and right inverses (Tag D-RT2FT)

A map of sets \(\pi\colon A\to B\) is a surjection if for every \(b\in B\) there exists \(a\in A\) with \(\pi(a)=b\). A right inverse of \(\pi\) is a map \(f\colon B\to A\) with \(\pi\circ f=\id_B\).

Separation axioms (Tag D-4DXA7)

Let \(X\) be a topological space. Two subsets \(S, T\subseteq X\) are separated by neighborhoods if there exist disjoint open sets \(U\supseteq S\) and \(V\supseteq T\).

• \(X\) is \(T_0\) if for all \(x\neq y\) in \(X\) there exists an open set containing exactly one of \(x\) and \(y\).

• \(X\) is \(T_1\) if for all \(x\neq y\) in \(X\) there exists an open set containing \(x\) and not \(y\).

• \(X\) is \(T_2\), or Hausdorff, if for all \(x\neq y\) in \(X\) the sets \(\ts{x}\) and \(\ts{y}\) are separated by neighborhoods.

• \(X\) is \(T_{2\frac12}\) if for all \(x\neq y\) in \(X\) there exist open sets \(U\ni x\) and \(V\ni y\) with \(\cl_X(U)\intersect\cl_X(V) = \emptyset\).

• \(X\) is \(T_3\) if it is \(T_0\) and regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\), the sets \(\ts{x}\) and \(F\) are separated by neighborhoods.

• \(X\) is \(T_{3\frac12}\) if it is \(T_0\) and completely regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\) there exists a continuous \(f\colon X\to[0,1]\) with \(f(x) = 0\) and \(f(F)\subseteq\ts{1}\).

• \(X\) is \(T_4\) if it is \(T_1\) and normal: any two disjoint closed subsets of \(X\) are separated by neighborhoods.

Topology via open sets and closed sets (Tag D-2TZAI)

Let \(X\) be a set. A topology on \(X\) is a collection \(\tau\) of subsets of \(X\) such that

• \(\emptyset \in \tau\) and \(X\in \tau\);

• \(\Union_{i\in I} U_i \in \tau\) for every family \((U_i)_{i\in I}\) of elements of \(\tau\);

• \(U_1\intersect \cdots \intersect U_n \in \tau\) for all \(n\geq 1\) and \(U_1, \ldots, U_n\in\tau\).

The pair \((X, \tau)\) is a topological space, the elements of \(\tau\) are its open sets, and a subset \(F\subseteq X\) is closed if \(X\sm F\) is open.

Separation axioms $T_0$ through $T_4$ (Tag D-OM7TD)

Let \(X\) be a topological space. Two subsets \(S, T\subseteq X\) are separated by neighborhoods if there exist disjoint open sets \(U\supseteq S\) and \(V\supseteq T\).

• \(X\) is \(T_0\) if for all \(x_1\neq x_2\) in \(X\) some open set contains exactly one of \(x_1,x_2\).

• \(X\) is \(T_1\) if for all \(x_1\neq x_2\) in \(X\) there is an open set containing \(x_1\) but not \(x_2\), and an open set containing \(x_2\) but not \(x_1\).

• \(X\) is \(T_2\) if any two distinct points are separated by neighborhoods; this is the Hausdorff condition.

• \(X\) is \(T_{2.5}\) if any two distinct points \(x_1,x_2\) have open neighborhoods \(U_1\ni x_1\) and \(U_2\ni x_2\) with \(\overline{U_1}\cap\overline{U_2}=\emptyset\).

• \(X\) is \(T_3\) if it is \(T_0\) and regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\), the sets \(\ts{x}\) and \(F\) are separated by neighborhoods.

• \(X\) is \(T_{3.5}\) if it is \(T_0\) and completely regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\) there is a continuous \(f\colon X\to[0,1]\) with \(f(x)=0\) and \(f(F)\subseteq\ts{1}\).

• \(X\) is \(T_4\) if it is \(T_1\) and normal: any two disjoint closed subsets of \(X\) are separated by neighborhoods.

Topological embedding (Tag D-BCNUH)

Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a topological embedding if it is injective and the map \(X\to f(X)\), \(x\mapsto f(x)\), is a homeomorphism, where \(f(X)\) carries the subspace topology.

Uniformly continuous map (Tag D-WGYSB)

Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. A map \(f\colon X\to Y\) is uniformly continuous if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all \(x_1, x_2\in X\), \[ d_X(x_1, x_2) < \delta \implies d_Y(f(x_1), f(x_2)) < \varepsilon. \]

Diameter of a subset of a metric space (Tag D-B7CYY)

Let \((X, d)\) be a metric space and \(A\subseteq X\) a nonempty subset. The diameter of \(A\) is \[ \diam(A) \coloneqq \sup_{p, q\in A}d(p, q) \in [0, \infty] .\]

Distance from a point to a subset (Tag D-D5G27)

Let \((X, d)\) be a metric space, \(S\subseteq X\) a nonempty subset, and \(x\in X\). The distance from \(x\) to \(S\) is \[ \dist(x, S)\coloneqq\inf_{s\in S} d(x, s). \]

Lebesgue number of an open cover (Tag D-SDMMS)

Let \((X,d)\) be a metric space and \(\mathcal U\) an open cover of \(X\). A real number \(\delta>0\) is a Lebesgue number for \(\mathcal U\) if every subset \(A\subseteq X\) with diameter \(\diam(A)<\delta\) is contained in some \(U\in\mathcal U\).

Totally bounded metric space (Tag FF-VWKGM)

A metric space \((M, d)\) is totally bounded if for every \(\eps > 0\) there are finitely many open balls of radius \(\eps\) whose union is \(M\) [@Mun00].

Basis of a module (Tag D-I7D56)

Let \(R\) be a ring and \(M\) a left $R$-module. A subset \(B\subseteq M\) is a basis of \(M\) if \(B\) is linearly independent and a generating set for \(M\).

Boundary of a manifold (Tag D-MLMIR)

Let \(n\geq 1\) and let \(M\) be a topological $n$-manifold with boundary. The boundary of \(M\) is the subspace \[ \del M \coloneqq \theset{x\in M \st H_n(M, M\setminus\theset{x}; \ZZ) = 0} \] [@Hat02].

Cap product (Tag D-RQS4J)

Let \(X\) be a topological space, \(R\) a commutative ring, and \(0\leq q\leq p\). For a singular \(p\)-simplex \(\sigma\colon\Delta^p=[v_0,\ldots,v_p]\to X\) and \(0\leq i\leq j\leq p\), write \(\sigma|_{[v_i,\ldots,v_j]}\) for the singular \((j-i)\)-simplex obtained by restricting \(\sigma\) to the face \([v_i,\ldots,v_j]\), identified with \(\Delta^{j-i}\) preserving the order of the vertices. The cap product on chains and cochains is the \(R\)-bilinear map \[ \frown\colon C_p(X;R)\times C^q(X;R)\to C_{p-q}(X;R),\qquad \sigma\frown\psi\coloneqq\psi\qty{\sigma|_{[v_0,\ldots,v_q]}}\,\sigma|_{[v_q,\ldots,v_p]} \] on singular \(p\)-simplices \(\sigma\) and singular $q$-cochains \(\psi\), extended linearly in \(\sigma\) [@Hat02].

Cellular homology (Tag D-A3PUW)

Let \(X\) be a CW complex with \(n\)-skeleta \(X^n\), and \(X^{-1}\coloneqq\emptyset\). The cellular chain complex of \(X\) has \(C_n^{\cell}(X) \coloneqq H_n(X^n, X^{n-1})\) and differential \(d_n\colon C_n^{\cell}(X)\to C_{n-1}^{\cell}(X)\) the composite \[ H_n(X^n, X^{n-1}) \mapsvia{\del_n} H_{n-1}(X^{n-1}) \mapsvia{j_{n-1}} H_{n-1}(X^{n-1}, X^{n-2}) \] of the boundary map of the long exact sequence of the pair \((X^n, X^{n-1})\) and the map induced by inclusion. The cellular homology \(H_n^{\cell}(X)\) is the homology of \((C_*^{\cell}(X), d_*)\).

Cellular map (Tag D-3IWX2)

Let \(X\) and \(Y\) be CW complexes with \(n\)-skeleta \(X^n\) and \(Y^n\). A continuous map \(f\colon X\to Y\) is cellular if \(f(X^n) \subseteq Y^n\) for every \(n\geq 0\).

Closed manifold (Tag D-SYKQW)

A closed manifold is a compact manifold without boundary [@Hat02].

Closed surface (Tag FD-QPIIL)

A closed surface is a compact \(2\)-dimensional manifold without boundary.

Constant map (Tag D-VZS33)

Let \(X\) and \(Y\) be sets. A map \(f\colon X\to Y\) is constant if there exists \(y_0\in Y\) such that \(f(x) = y_0\) for every \(x\in X\).

Colimit of a directed system (Tag D-TD6AO)

Let \(\mathcal C\) be a category and \((I,\leq)\) a directed set. A directed system in \(\mathcal C\) consists of objects \(X_i\) for \(i\in I\) and morphisms \(f_{ij}\colon X_i\to X_j\) for \(i\leq j\), with \(f_{ii}=\id_{X_i}\) and \(f_{jk}\circ f_{ij}=f_{ik}\) for \(i\leq j\leq k\). A colimit of the system is an object \(X\) with morphisms \(\iota_i\colon X_i\to X\) satisfying \(\iota_j\circ f_{ij}=\iota_i\) for \(i\leq j\), such that for every object \(Y\) and morphisms \(\psi_i\colon X_i\to Y\) with \(\psi_j\circ f_{ij}=\psi_i\) for \(i\leq j\) there is a unique morphism \(h\colon X\to Y\) with \(h\circ\iota_i=\psi_i\) for all \(i\).

Colimit (Tag D-5MX7E)

Let \(\mathcal C\) be a category, \(I\) a small category, and \(F\colon I\to\mathcal C\) a functor. A cocone on \(F\) is an object \(Y\) of \(\mathcal C\) with morphisms \(\psi_i\colon F(i)\to Y\) for \(i\in I\) such that \(\psi_j\circ F(u) = \psi_i\) for every morphism \(u\colon i\to j\) in \(I\). A colimit of \(F\) is a cocone \((X, (\iota_i)_{i\in I})\) such that for every cocone \((Y, (\psi_i)_{i\in I})\) there exists a unique morphism \(\varphi\colon X\to Y\) with \(\varphi\circ\iota_i = \psi_i\) for all \(i\in I\).

Contractible space (Tag D-K43GA)

A topological space \(X\) is contractible if \(\id_X\) is nullhomotopic: there is a point \(x_0\in X\) such that \(\id_X\) is homotopic to the constant map \(c\colon X\to X\), \(c(x) = x_0\).

Coproduct (Tag D-COC6C)

Let \(\mathcal C\) be a category and \((X_\alpha)_{\alpha\in A}\) a family of objects of \(\mathcal C\). A coproduct of \((X_\alpha)_{\alpha\in A}\) is an object \(\coprod_{\alpha\in A} X_\alpha\) together with morphisms \(\iota_\beta\colon X_\beta\to\coprod_{\alpha\in A} X_\alpha\) for \(\beta\in A\) such that for every object \(Y\) and every family of morphisms \(f_\beta\colon X_\beta\to Y\) there is a unique morphism \(f\colon\coprod_{\alpha\in A} X_\alpha\to Y\) with \(f\circ\iota_\beta = f_\beta\) for every \(\beta\in A\).

Covering space (Tag D-ANO2D)

Let \(X\) be a topological space. A covering space of \(X\) is a topological space \(\tilde X\) together with a continuous map \(p\colon \tilde X\to X\) such that every \(x\in X\) has an open neighborhood \(U\) for which \(\inverseof{p}(U)\) is a union of pairwise disjoint open sets \(V_i\subseteq\tilde X\), each mapped homeomorphically onto \(U\) by \(p\). The sets \(V_i\) are the sheets of \(\tilde X\) over \(U\).

Cup product (Tag D-B2JER)

Let \(X\) be a topological space, \(R\) a ring, and \(p, q\geq 0\). For singular cochains \(\varphi\in C^p(X;R)\) and \(\psi\in C^q(X;R)\), the cup product \(\varphi\smile\psi\in C^{p+q}(X;R)\) is the cochain whose value on a singular \((p+q)\)-simplex \(\sigma\colon[v_0, \ldots, v_{p+q}]\to X\) is \[ (\varphi\smile\psi)(\sigma) \coloneqq \varphi\qty{\ro{\sigma}{[v_0, \ldots, v_p]}}\,\psi\qty{\ro{\sigma}{[v_p, \ldots, v_{p+q}]}} ,\] where each face is identified with the standard simplex of its dimension by the order-preserving linear homeomorphism.

CW complex (Tag D-ZOU5G)

A CW complex is a topological space \(X\) together with subspaces \(X^0\subseteq X^1\subseteq\cdots\) constructed as follows.

1. \(X^0\) is a discrete space; its points are the \(0\)-cells.

2. For \(n\geq 1\), the \(n\)-skeleton \(X^n\) is obtained from \(X^{n-1}\) by attaching a family of \(n\)-cells \(e^n_\alpha\) along continuous maps \(\varphi_\alpha\colon S^{n-1}\to X^{n-1}\): \[ X^n = \qty{X^{n-1} \disjoint \Disjoint_\alpha D^n_\alpha} / \qty{x \sim \varphi_\alpha(x) \text{ for } x \in \del D^n_\alpha}, \] with the quotient topology.

3. Either \(X = X^n\) for some \(n\), or \(X = \Union_{n\geq 0} X^n\) with the weak topology: a subset \(A\subseteq X\) is closed if and only if \(A \intersect X^n\) is closed in \(X^n\) for every \(n\).

Attaching an $n$-cell (Tag D-MMDM3)

Let \(X\) be a topological space and \(n\geq 0\). Let \(D^n\subseteq\RR^n\) be the closed unit ball and \(S^{n-1}=\del D^n\) its boundary sphere, with \(S^{-1}=\emptyset\). For a continuous map \(f\colon S^{n-1}\to X\), the space obtained by attaching an \(n\)-cell to \(X\) along \(f\) is the quotient space \[ X\cup_f D^n \coloneqq \qty{X\disjoint D^n}/\qty{s\sim f(s) \text{ for } s\in S^{n-1}}, \] the case \(X_0=X\), \(X_1=D^n\), \(A=S^{n-1}\) of attaching a space along a map. The map \(f\) is the attaching map. The composite \(\Phi\colon D^n\to X\disjoint D^n\to X\cup_f D^n\) of the inclusion and the quotient map is the characteristic map, its image \(\Phi(D^n)\) is the closed \(n\)-cell, and \(e^n\coloneqq\Phi(D^n\setminus S^{n-1})\) is the open \(n\)-cell.

Deck transformation (Tag D-SI6OM)

Let \(p\colon\tilde X\to X\) be a covering space. A deck transformation of \(p\) is a homeomorphism \(\psi\colon\tilde X\to\tilde X\) with \(p\circ\psi=p\). The deck transformations form a group \(\Deck(\tilde X\to X)\) under composition.

Deformation and deformation retraction (Tag D-UH3L5)

Let \(X\) be a topological space and \(I=[0,1]\). A deformation of \(X\) is a family of maps \(f_t\colon X\to X\), \(t\in I\), such that \(f_0=\id_X\) and \((x,t)\mapsto f_t(x)\) is a continuous map \(X\times I\to X\). For a subspace \(A\subseteq X\), a deformation retraction of \(X\) onto \(A\) is a deformation \((f_t)_{t\in I}\) of \(X\) with \(f_1(X)=A\) and \(f_t(a)=a\) for all \(a\in A\) and \(t\in I\) [@Hat02].

Deformation retraction (Tag D-6UHU7)

Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A deformation retraction of \(X\) onto \(A\) is a homotopy \(F\colon X\cross I\to X\), written \(F_t \coloneqq F(\wait, t)\), such that for all \(t\in I\): \[ F_0 = \id_X, \qquad F_1(X) \subseteq A, \qquad F_t(a) = a \text{ for all } a\in A .\] The subspace \(A\) is a deformation retract of \(X\) if there exists a deformation retraction of \(X\) onto \(A\).

Degree of a map of spheres (Tag D-XC53X)

Let \(n\geq 1\) and let \(f\colon S^n \to S^n\) be a continuous map. Since \(H^n(S^n;\ZZ) \cong \ZZ\), the induced homomorphism \(f^*\colon H^n(S^n;\ZZ) \to H^n(S^n;\ZZ)\) is multiplication by a unique integer \(d\), that is, \(f^*(u) = d\,u\) for every \(u\in H^n(S^n;\ZZ)\). The degree of \(f\) is \(\deg f \coloneqq d\).

Euler characteristic of a finite CW complex (Tag D-QK5BM)

Let \(X\) be a finite CW complex with \(c_n\) cells of dimension \(n\). The Euler characteristic of \(X\) is \[ \chi(X)\coloneqq\sum_{n\geq0}(-1)^nc_n. \]

Left exact, right exact, and exact functors (Tag D-S7L6M)

Let \(T\colon\mathcal A\to\mathcal B\) be an additive covariant functor between abelian categories, such as categories of modules. The functor \(T\) is right exact if for every short exact sequence \(0\to A\mapsvia{f}B\mapsvia{g}C\to0\) in \(\mathcal A\) the sequence \[ T(A)\mapsvia{T(f)}T(B)\mapsvia{T(g)}T(C)\to0 \] is exact, and left exact if for every such sequence \[ 0\to T(A)\mapsvia{T(f)}T(B)\mapsvia{T(g)}T(C) \] is exact. It is exact if it is both left and right exact, that is, if it takes every short exact sequence to a short exact sequence.

Excision axiom (Tag D-455S6)

Let \(h_*\) be a sequence of functors \(h_n\) from pairs of topological spaces to abelian groups. The functors \(h_*\) satisfy excision if for every pair \((X, A)\) and every subspace \(Z\subseteq A\) with \(\cl_X(Z) \subseteq A^\circ\), the inclusion \((X\sm Z, A\sm Z)\injects (X, A)\) induces isomorphisms \[ h_n(X\sm Z, A\sm Z) \mapsvia{\sim} h_n(X, A) \quad\text{for all } n .\]

Free and properly discontinuous group actions (Tag D-2PNEG)

Let \(G\) be a group acting on a topological space \(X\) by homeomorphisms, \(G\actson X\).

• The action is free if \(g(x) = x\) for some \(x\in X\) implies \(g = e\).

• The action is properly discontinuous if every \(x\in X\) has a neighborhood \(U\) such that for all \(g_1, g_2\in G\), \(g_1(U) \intersect g_2(U) \neq \emptyset\) implies \(g_1 = g_2\).

Free product of groups (Tag D-JDDCP)

Let \((G_\alpha)_{\alpha\in A}\) be a family of groups. A reduced word is a finite sequence \(g_1g_2\cdots g_k\), \(k\geq 0\), in which each letter \(g_i\) lies in \(G_{\alpha_i}\sm\ts{1}\) for some \(\alpha_i\in A\) and \(\alpha_i\neq\alpha_{i+1}\) for \(1\leq i<k\). The free product \(\ast_{\alpha\in A} G_\alpha\) is the set of reduced words, with product given by concatenating two words and then reducing: adjacent letters from the same \(G_\alpha\) are replaced by their product in \(G_\alpha\), and letters equal to \(1\) are deleted, until the word is reduced.

Free product with amalgamation (Tag D-WSFYS)

Let \(A\), \(B\), and \(C\) be groups, and let \(i\colon C\to A\) and \(j\colon C\to B\) be homomorphisms. The free product with amalgamation of \(A\) and \(B\) over \(C\) is \[ A \ast_C B \coloneqq \qty{A \ast B} / N, \] where \(A\ast B\) is the free product and \(N\) is the normal subgroup of \(A\ast B\) generated by \(\ts{ i(c) \inverseof{j(c)} \st c\in C }\).

Fundamental class (Tag D-TS7TZ)

Let \(M\) be a closed connected \(n\)-manifold with an orientation \(x\mapsto\mu_x\in H_n(M,M\setminus\ts{x};\ZZ)\). A fundamental class of \(M\) is a class \([M]\in H_n(M;\ZZ)\) whose image under \(H_n(M;\ZZ)\to H_n(M,M\setminus\ts{x};\ZZ)\) is \(\mu_x\) for every \(x\in M\).

Fundamental group (Tag D-EBNUE)

Let \(X\) be a topological space and \(x_0\in X\), and let \(I = [0,1]\). The fundamental group \(\pi_1(X, x_0)\) is the set of classes of loops \(\gamma\colon I\to X\) based at \(x_0\) under homotopy rel endpoints, with multiplication \[ [\gamma][\eta]\coloneqq[\gamma\cdot\eta], \] where \(\gamma\cdot\eta\) is the concatenation of \(\gamma\) and \(\eta\).

Attaching a space along a map (Tag D-HRU62)

Let \(X_0\) and \(X_1\) be topological spaces, \(A\subseteq X_1\) a subspace, and \(f\colon A\to X_0\) a continuous map. The space obtained by attaching \(X_1\) to \(X_0\) along \(f\) is the quotient space \[ X_0\sqcup_f X_1\coloneqq\qty{X_0\sqcup X_1}/\qty{a\sim f(a) \text{ for } a\in A}, \] where \(\sim\) is the equivalence relation generated by \(a\sim f(a)\).

Homotopic maps (Tag D-Z7I7F)

Let \(X\) and \(Y\) be topological spaces, \(I = [0,1]\), and \(f, g\colon X\to Y\) continuous maps. The maps \(f\) and \(g\) are homotopic, written \(f\homotopic g\), if there is a continuous map \(H\colon X\cross I \to Y\) with \(H(x, 0) = f(x)\) and \(H(x, 1) = g(x)\) for all \(x\in X\); such a map \(H\) is a homotopy from \(f\) to \(g\). Two paths \(\gamma, \eta\colon I\to X\) with \(\gamma(0) = \eta(0)\) and \(\gamma(1) = \eta(1)\) are homotopic rel endpoints if there is a continuous map \(H\colon I\cross I\to X\) with \(H(s, 0) = \gamma(s)\), \(H(s, 1) = \eta(s)\), \(H(0,t) = \gamma(0)\), and \(H(1,t) = \gamma(1)\) for all \(s, t\in I\).

Homotopy class (Tag D-IZI3T)

Let \(X\) and \(Y\) be topological spaces. The homotopy class \([f]\) of a continuous map \(f\colon X\to Y\) is its equivalence class under the relation \(\simeq\) of being homotopic. The set of homotopy classes of continuous maps \(X\to Y\) is \[ [X, Y]\coloneqq\Hom_{\Top}(X, Y)/\simeq. \]

Homotopy equivalence (Tag D-HFR32)

Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a homotopy equivalence if there is a continuous map \(g\colon Y\to X\) with \(f\circ g\simeq\id_Y\) and \(g\circ f\simeq\id_X\), where \(\simeq\) denotes homotopy. Such a map \(g\) is a homotopy inverse of \(f\). The spaces \(X\) and \(Y\) are homotopy equivalent, or have the same homotopy type, written \(X\simeq Y\), if there is a homotopy equivalence \(X\to Y\).

Homotopy extension property (Tag D-KVAI3)

Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(I = [0,1]\). The pair \((X, A)\) has the homotopy extension property if for every space \(Y\), every continuous map \(f_0\colon X\to Y\), and every homotopy \(g_t\colon A\to Y\) with \(g_0 = f_0|_A\), there is a homotopy \(f_t\colon X\to Y\) starting at \(f_0\) with \(f_t|_A = g_t\) for all \(t\in I\). Equivalently, every pair of continuous maps \(X\times\ts{0}\to Y\) and \(A\times I\to Y\) that agree on \(A\times\ts{0}\) extends to a continuous map \(X\times I\to Y\).

Homotopy groups (Tag D-EUX36)

Let \(X\) be a topological space, \(x_0\in X\), and \(n\geq 1\), and let \(I^n = [0,1]^n\) with boundary \(\del I^n\). The \(n\)th homotopy group \(\pi_n(X, x_0)\) is the set of homotopy classes of maps \(f\colon(I^n, \del I^n)\to(X, x_0)\), where the homotopies \(f_t\) are required to satisfy \(f_t(\del I^n) = \ts{x_0}\) for all \(t\), with the operation \([f]+[g]\coloneqq[f+g]\) given by \[ (f+g)(s_1, s_2, \ldots, s_n)\coloneqq \begin{cases} f(2s_1, s_2, \ldots, s_n) & 0\leq s_1\leq 1/2, \\ g(2s_1-1, s_2, \ldots, s_n) & 1/2\leq s_1\leq 1. \end{cases} \]

Homotopy lifting property (Tag D-HOCNK)

Let \(p\colon E\to B\) be a continuous map and \(Y\) a topological space. The map \(p\) has the homotopy lifting property with respect to \(Y\) if for every homotopy \(g_t\colon Y\to B\), \(t\in[0,1]\), and every continuous \(\tilde g_0\colon Y\to E\) with \(p\circ\tilde g_0 = g_0\), there is a homotopy \(\tilde g_t\colon Y\to E\) starting at \(\tilde g_0\) with \(p\circ\tilde g_t = g_t\) for all \(t\). The map \(p\) is a fibration if it has the homotopy lifting property with respect to every space \(Y\).

Intersection product and intersection pairing (Tag D-TK4QD)

Let \(M\) be a closed connected oriented \(n\)-manifold with fundamental class \([M]\in H_n(M;\ZZ)\), and let \(D\colon H^k(M;\ZZ)\to H_{n-k}(M;\ZZ)\), \(D(\varphi)=[M]\frown\varphi\), be the Poincaré duality isomorphism. The intersection product is the bilinear map \[ H_i(M;\ZZ)\times H_j(M;\ZZ)\to H_{i+j-n}(M;\ZZ),\qquad \alpha\cdot\beta\coloneqq[M]\frown\qty{\inverseof{D}(\alpha)\smile \inverseof{D}(\beta)}, \] using the cup product and the cap product. For \(j=n-i\), composing with \(H_0(M;\ZZ)\cong\ZZ\) gives the intersection pairing \(H_i(M;\ZZ)\times H_{n-i}(M;\ZZ)\to\ZZ\).

Intersection form of a manifold (Tag D-6POU4)

Let \(M\) be a closed, connected, oriented \(2k\)-manifold with fundamental class \([M]\in H_{2k}(M;\ZZ)\). The intersection form of \(M\) is the bilinear form \[ I\colon H^k(M;\ZZ)\cross H^k(M;\ZZ) \to \ZZ, \qquad I(a, b)\coloneqq \inner{a\smile b}{[M]} ,\] where \(\smile\) is the cup product and \(\inner{\wait}{\wait}\) is the Kronecker pairing.

Kronecker pairing (Tag D-VP4LC)

Let \(X\) be a topological space, \(R\) a commutative ring, and \(n\geq 0\). For a singular cochain \(\varphi\in C^n(X;R) = \Hom_R(C_n(X;R), R)\) and a singular chain \(\alpha\in C_n(X;R)\), put \(\inner{\varphi}{\alpha} \coloneqq \varphi(\alpha)\). The Kronecker pairing, also called the Kronecker product, is the \(R\)-bilinear map \[ \inner{\wait}{\wait}\colon H^n(X; R) \cross H_n(X; R) \to R, \qquad \inner{[\varphi]}{[\alpha]} \coloneqq \varphi(\alpha), \] for a cocycle \(\varphi\) and a cycle \(\alpha\). Its adjoint is the evaluation homomorphism \[ h\colon H^n(X; R) \to \Hom_R(H_n(X;R), R), \qquad h([\varphi])([\alpha]) \coloneqq \varphi(\alpha). \]

Lefschetz number (Tag D-3UY5O)

Let \(X\) be a finite simplicial complex, or a retract of one, and let \(f\colon X\to X\) be a continuous map. The Lefschetz number of \(f\) is \[ \tau(f) \coloneqq \sum_{n\geq 0} (-1)^n \tr\qty{ f_*\colon H_n(X;\QQ) \to H_n(X;\QQ) } .\]

Lens space (Tag D-UI3FE)

Let \(m\geq 2\), \(n \geq 1\) and let \(\ell_1, \ldots, \ell_n\) be integers coprime to \(m\). Let \(\ZZ/m\) act on the unit sphere \(S^{2n-1}\subseteq \CC^n\) with the generator acting by \[ (z_1, \ldots, z_n) \mapsto \qty{ e^{2\pi i \ell_1/m} z_1, \ldots, e^{2\pi i \ell_n /m} z_n } .\] The lens space is the orbit space \[ L_m(\ell_1, \ldots, \ell_n) \coloneqq S^{2n-1}/\qty{\ZZ/m} \] [@Hat02].

Local degree (Tag D-ZVY6X)

Let \(n\geq 1\), let \(f\colon S^n\to S^n\) be a continuous map, and let \(y\in S^n\) be a point whose preimage \(\inverseof{f}(y) = \ts{x_1, \ldots, x_m}\) is finite. Homology is taken with coefficients in \(\ZZ\). Choose an open neighborhood \(V\) of \(y\) and disjoint open neighborhoods \(U_i\) of \(x_i\) with \(f(U_i)\subseteq V\), and fix a generator of \(H_n(S^n)\). Then \(f\) restricts to maps of pairs \(f_i\colon (U_i, U_i\sm\ts{x_i})\to (V, V\sm\ts{y})\), and under the isomorphisms \[ H_n(U_i, U_i\sm\ts{x_i})\cong H_n(S^n, S^n\sm\ts{x_i})\cong H_n(S^n)\cong\ZZ, \qquad H_n(V, V\sm\ts{y})\cong H_n(S^n, S^n\sm\ts{y})\cong H_n(S^n)\cong\ZZ \] given by excision and the long exact sequence of the pair, the induced homomorphism \(f_{i*}\) is multiplication by an integer. This integer is the local degree \(\deg f\vert_{x_i}\) of \(f\) at \(x_i\).

Local orientation and orientation of a manifold (Tag D-6CI7D)

Let \(M\) be an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A)\coloneqq H_n(M, M\sm A;\ZZ)\). For \(x\in M\), excision gives \(H_n(M\mid x)\cong H_n(\RR^n, \RR^n\sm\ts{0};\ZZ)\cong\ZZ\).

• A local orientation of \(M\) at \(x\) is a generator \(\mu_x\) of \(H_n(M\mid x)\).

• An orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a local orientation at \(x\), such that each \(x\in M\) has an open neighborhood \(U\subseteq M\) with a homeomorphism \(U\cong\RR^n\) and an open ball \(B\subseteq U\) of finite radius containing \(x\), together with a generator \(\mu_B\) of \(H_n(M\mid B)\cong\ZZ\) whose image under the map \(H_n(M\mid B)\to H_n(M\mid y)\) induced by inclusion is \(\mu_y\) for every \(y\in B\).

• \(M\) is orientable if an orientation of \(M\) exists.

Local homology (Tag D-BNCTG)

Let \(X\) be a topological space, \(A\subseteq X\) a subset, and \(n\geq 0\). The local homology of \(X\) at \(A\) in degree \(n\) is the relative homology group \[ H_n(X\mid A) \coloneqq H_n(X, X\sm A; \ZZ) .\] For a point \(x\in X\), write \(H_n(X\mid x)\coloneqq H_n(X\mid\ts{x})\).

Manifold (Tag D-UBWVX)

Let \(n\geq0\). An \(n\)-manifold is a Hausdorff space \(M\) in which every point has an open neighborhood homeomorphic to \(\RR^n\) [@Hat02].

Manifold with boundary (Tag D-MN6QW)

Let \(n\geq 1\) and let \(\mathbb H^n\coloneqq\ts{x\in\RR^n \st x_n\geq 0}\) be the closed upper half-space, with the subspace topology. A topological \(n\)-manifold with boundary is a Hausdorff space \(M\) in which every point has an open neighborhood homeomorphic to \(\RR^n\) or to \(\mathbb H^n\).

Mayer--Vietoris sequence (Tag D-FAIJX)

Let \(X\) be a topological space and \(A, B\subseteq X\) subspaces with \(X = A^\circ\cup B^\circ\). Let \(i\colon A\cap B\hookrightarrow A\), \(j\colon A\cap B\hookrightarrow B\), \(k\colon A\hookrightarrow X\), and \(l\colon B\hookrightarrow X\) be the inclusions, and let \(H_n\) denote singular homology with integer coefficients. Define \[ \Phi\colon H_n(A\cap B)\to H_n(A)\oplus H_n(B),\quad \Phi(x) = (i_*x, -j_*x), \qquad \Psi\colon H_n(A)\oplus H_n(B)\to H_n(X),\quad \Psi(a, b) = k_*a + l_*b, \] and \(\del\colon H_n(X)\to H_{n-1}(A\cap B)\) as follows: every \(\alpha\in H_n(X)\) is represented by a cycle \(z = x + y\) with \(x\) an \(n\)-chain in \(A\) and \(y\) an \(n\)-chain in \(B\), and \(\del\alpha\coloneqq[\del x]\). The Mayer–Vietoris sequence of \((A, B)\) is \[ \cdots\to H_n(A\cap B)\xrightarrow{\Phi} H_n(A)\oplus H_n(B)\xrightarrow{\Psi} H_n(X)\xrightarrow{\del} H_{n-1}(A\cap B)\to\cdots\to H_0(X)\to 0. \]

Monodromy action on a fiber (Tag D-6BK54)

Let \(p\colon \tilde X \to X\) be a covering space and \(x_0 \in X\). For \(\tilde x\in \inverseof{p}(x_0)\) and a loop \(\gamma\) in \(X\) based at \(x_0\), let \(\tilde\gamma\) be the unique lift of \(\gamma\) with \(\tilde\gamma(0) = \tilde x\). The monodromy action is the right action of \(\pi_1(X, x_0)\) on \(\inverseof{p}(x_0)\) given by \[ \tilde x \cdot [\gamma] \coloneqq \tilde\gamma(1) .\]

Cells of a CW complex (Tag D-MEPE3)

Let \(X\) be a CW complex whose \(n\)-skeleton \(X^n\) is obtained from \(X^{n-1}\) by attaching \(n\)-disks \(D^n_\alpha\) along attaching maps \(\varphi_\alpha\colon S^{n-1}\to X^{n-1}\). The characteristic map of \(D^n_\alpha\) is the composite \(\Phi_\alpha\colon D^n_\alpha\hookrightarrow X^{n-1}\sqcup\coprod_\beta D^n_\beta\to X^n\hookrightarrow X\), whose restriction to \(S^{n-1}\) is \(\varphi_\alpha\). The \(n\)-cell \(e^n_\alpha\) is the image \(\Phi_\alpha(D^n\sm S^{n-1})\).

$n$-connected spaces and pairs (Tag D-4QNEW)

Let \(n\geq 0\). A nonempty topological space \(X\) is \(n\)-connected if \(X\) is path-connected and \(\pi_i(X, x_0) = 0\) for all \(1\leq i\leq n\) and all \(x_0\in X\). A pair \((X, A)\) with \(A\subseteq X\) is \(n\)-connected if every path component of \(X\) meets \(A\) and the relative homotopy groups satisfy \(\pi_i(X, A, x_0) = 0\) for all \(1\leq i\leq n\) and all \(x_0\in A\).

Galois, normal, and regular covering spaces (Tag D-MWD2L)

Let \(p\colon\tilde X\to X\) be a covering space, let \(x_0\in X\), and let \(\Deck(\tilde X)\) be its group of deck transformations. The covering \(p\) is Galois, also called normal or regular, over \(x_0\) if \(\Deck(\tilde X)\) acts transitively on the fiber \(\inverseof{p}(x_0)\): for all \(\tilde x_1,\tilde x_2\in \inverseof{p}(x_0)\) there is \(\psi\in\Deck(\tilde X)\) with \(\psi(\tilde x_1)=\tilde x_2\).

Nullhomotopic map (Tag D-MGRZP)

Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is nullhomotopic if it is homotopic to a constant map: there are \(y_0\in Y\) and a continuous map \(F\colon X\times[0,1]\to Y\) with \(F(x, 0) = f(x)\) and \(F(x, 1) = y_0\) for all \(x\in X\).

Orbit space (Tag D-O5NN7)

Let \(G\) be a group acting on a topological space \(X\). For \(x,y\in X\) write \(x\sim y\) if there exists \(g\in G\) with \(g\cdot x=y\); this is an equivalence relation whose classes are the orbits of the action. Let \(X/G\coloneqq X/{\sim}\) be the set of orbits and \(q\colon X\to X/G\) the map sending \(x\) to its orbit. The orbit space is \(X/G\) with the quotient topology: \(U\subseteq X/G\) is open if and only if \(\inverseof{q}(U)\) is open in \(X\), so that \(q\) is a quotient map.

Orientable manifold (Tag D-K5MLW)

An \(n\)-manifold \(M\) is orientable if there exists an orientation of \(M\).

Orientation cover (Tag D-WX7JH)

Let \(M\) be an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A) \coloneqq H_n(M, M\sm A;\ZZ)\). The orientation double cover of \(M\) is the set \(\tilde M\) of all local orientations \(\mu_x\) of \(M\) at points \(x\in M\), with the map \(p\colon\tilde M\to M\), \(\mu_x\mapsto x\), and the topology with basis the sets \[ U(\mu_B) \coloneqq \ts{ \mu_x \st x\in B,\ \mu_x \text{ the image of } \mu_B \text{ under } H_n(M\mid B)\to H_n(M\mid x) }, \] where \(B\) ranges over the open balls of finite radius in charts \(\RR^n\subseteq M\) and \(\mu_B\) over the generators of \(H_n(M\mid B)\cong\ZZ\).

Oriented manifold (Tag D-CNLBT)

Let \(M\) be an \(n\)-manifold. An orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a local orientation \(\mu_x\), a generator of \(H_n(M, M\sm\ts{x};\ZZ)\cong\ZZ\), such that every \(x\in M\) has an open ball neighborhood \(B\) of finite radius in a chart \(\RR^n\subseteq M\) and a class \(\mu_B\in H_n(M, M\sm B;\ZZ)\) whose image under the map \(H_n(M, M\sm B;\ZZ)\to H_n(M, M\sm\ts{y};\ZZ)\) induced by inclusion is \(\mu_y\) for every \(y\in B\). An oriented manifold is a pair consisting of a manifold and an orientation of it. The manifold \(M\) is orientable if an orientation of \(M\) exists.

Path (Tag D-J6XOC)

Let \(X\) be a topological space and \(I\coloneqq[0,1]\). A path in \(X\) is a continuous map \(\gamma\colon I\to X\); it is a path from \(\gamma(0)\) to \(\gamma(1)\). A path \(\gamma\) is a loop if \(\gamma(0) = \gamma(1)\), and then it is a loop based at \(\gamma(0)\). For paths \(\gamma, \eta\colon I\to X\) with \(\gamma(1) = \eta(0)\), the concatenation \(\gamma\cdot\eta\colon I\to X\) is the path \[ (\gamma\cdot\eta)(s)\coloneqq \begin{cases} \gamma(2s) & 0\leq s\leq 1/2, \\ \eta(2s-1) & 1/2\leq s\leq 1. \end{cases} \]

Path lifting property (Tag D-OISBB)

Let \(p\colon E\to B\) be a continuous map and \(I=[0,1]\). The map \(p\) has the path lifting property if for every path \(\gamma\colon I\to B\) and every \(e_0\in \inverseof{p}(\gamma(0))\) there is a path \(\tilde\gamma\colon I\to E\) with \(p\circ\tilde\gamma=\gamma\) and \(\tilde\gamma(0)=e_0\).

Poincaré duality (Tag D-QP7WI)

Let \(R\) be a commutative ring and \(M\) a closed $R$-orientable \(n\)-manifold with fundamental class \([M]\in H_n(M;R)\). The Poincaré duality map is \[ D\colon H^k(M;R)\to H_{n-k}(M;R),\qquad D(\alpha)\coloneqq[M]\frown\alpha, \] given by the cap product with \([M]\).

$R$-orientability (Tag D-YD6DR)

Let \(R\) be a commutative ring with identity and \(M\) an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A;R)\coloneqq H_n(M, M\sm A; R)\), so that \(H_n(M\mid x;R)\cong R\) for \(x\in M\). An \(R\)-orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a generator \(\mu_x\) of the \(R\)-module \(H_n(M\mid x;R)\) that is locally consistent: each \(x\in M\) has a neighborhood \(\RR^n\subseteq M\) containing an open ball \(B\) of finite radius about \(x\) and an element \(\mu_B\in H_n(M\mid B;R)\) whose image under the map \(H_n(M\mid B;R)\to H_n(M\mid y;R)\) induced by inclusion is \(\mu_y\) for every \(y\in B\). The manifold \(M\) is \(R\)-orientable if it has an \(R\)-orientation.

Relative boundaries (Tag D-Y6LXB)

Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(n\geq 0\), and regard the singular chain group \(C_n(A)\) as a subgroup of \(C_n(X)\) via the inclusion. A relative boundary is a chain \(\alpha\in C_n(X)\) of the form \(\alpha = \del\beta + \gamma\) with \(\beta \in C_{n+1}(X)\) and \(\gamma \in C_n(A)\).

Relative cycle (Tag D-2XKM5)

Let \(X\) be a topological space and \(A\subseteq X\) a subspace, and regard the singular chain group \(C_n(A)\) as a subgroup of \(C_n(X)\) via the inclusion. The group of relative chains is \(C_n(X,A) \coloneqq C_n(X)/C_n(A)\). A relative cycle is a chain \(\alpha \in C_n(X)\) with \(\del_n \alpha \in C_{n-1}(A)\).

Relative homotopy groups (Tag D-VUDRJ)

Let \(X\) be a topological space, \(A\subseteq X\) a subspace, \(x_0 \in A\), and \(n\geq 1\). Let \(D^n\) be the closed unit ball, \(S^{n-1} = \del D^n\), and \(s_0\in S^{n-1}\) a basepoint. A map of triples \(f\colon (D^n, S^{n-1}, s_0) \to (X, A, x_0)\) is a continuous map \(f\colon D^n\to X\) with \(f(S^{n-1})\subseteq A\) and \(f(s_0) = x_0\). The relative homotopy group is \[ \pi_n(X, A, x_0) \coloneqq \ts{ f\colon (D^n, S^{n-1}, s_0) \to (X, A, x_0) } / \homotopic, \] the set of homotopy classes of maps of triples, where the homotopies are through maps of triples.

Semilocally simply connected space (Tag D-EPQ54)

A topological space \(X\) is semilocally simply connected if every \(x\in X\) has a neighborhood \(U\) such that the homomorphism \(\pi_1(U, x)\to\pi_1(X, x)\) of fundamental groups induced by the inclusion \(U\hookrightarrow X\) is trivial.

Short exact sequence (Tag D-SIUWU)

Let \(R\) be a ring. A short exact sequence of \(R\)-modules is an exact sequence \[ 0\to A\mapsvia{i}B\mapsvia{j}C\to0 \] of \(R\)-modules and \(R\)-linear maps; equivalently, \(i\) is injective, \(j\) is surjective, and \(\im i=\ker j\) [@Hat02]. It splits if there is an \(R\)-linear map \(s\colon C\to B\) with \(j\circ s=\id_C\).

Simplicial complex (Tag D-NJ2Y6)

Let \(N\geq0\) and let \(v_0,\ldots,v_k\in\RR^N\) be affinely independent, meaning that \(v_1-v_0,\ldots,v_k-v_0\) are linearly independent. The \(k\)-simplex \([v_0,\ldots,v_k]\subseteq\RR^N\) is the convex hull of \(v_0,\ldots,v_k\), and these points are its vertices. A face of \([v_0,\ldots,v_k]\) is a simplex spanned by a nonempty subset of \(\ts{v_0,\ldots,v_k}\); for \(0\leq i\leq k\) and \(k\geq1\), the \(i\)-th facet is \[ \del_i[v_0,\ldots,v_k]\coloneqq[v_0,\ldots,\hat v_i,\ldots,v_k]. \] A simplicial complex in \(\RR^N\) is a set \(K\) of simplices in \(\RR^N\) such that:

1. every face of a simplex in \(K\) belongs to \(K\);

2. for \(\sigma,\tau\in K\), the intersection \(\sigma\cap\tau\) is either empty or a face of both \(\sigma\) and \(\tau\);

3. every point of \(\RR^N\) has a neighborhood that meets only finitely many simplices of \(K\).

Simplicial map (Tag D-RU2GC)

Let \(K\) and \(L\) be simplicial complexes with vertex sets \(V(K)\) and \(V(L)\). A simplicial map \(f\colon K\to L\) is a map \(f\colon V(K)\to V(L)\) such that whenever \(v_0,\ldots,v_k\) are the vertices of a simplex of \(K\), the points \(f(v_0),\ldots,f(v_k)\), not necessarily distinct, are the vertices of a simplex of \(L\).

Simply connected space (Tag D-GFM35)

A topological space \(X\) is simply connected if \(X\) is path connected and the fundamental group \(\pi_1(X, x_0)\) is trivial for every \(x_0\in X\).

Singular chain (Tag D-M3Y6X)

Let \(X\) be a topological space, \(R\) a commutative ring, and \(p\geq 0\). A singular \(p\)-chain in \(X\) with coefficients in \(R\) is an element of the free \(R\)-module \(C_p(X;R)\) on the singular \(p\)-simplices \(\sigma\colon\Delta^p\to X\), that is, a finite sum \(c = \sum_{i=1}^k r_i\sigma_i\) with \(r_i\in R\) and \(\sigma_i\colon\Delta^p\to X\) continuous.

Singular cochain (Tag D-R6LA3)

Let \(X\) be a topological space, \(G\) an abelian group, and \(n\geq0\). Let \(C_n(X)\) be the free abelian group on the singular \(n\)-simplices \(\sigma\colon\Delta^n\to X\), as in singular homology. A singular \(n\)-cochain of \(X\) with coefficients in \(G\) is a homomorphism \(\varphi\colon C_n(X)\to G\), and \[ C^n(X;G)\coloneqq\Hom(C_n(X),G) \] is the group of singular \(n\)-cochains [@Hat02].

Singular homology (Tag D-6BUWA)

Let \(X\) be a topological space and \(n\geq 0\), and let \(\Delta^n = [v_0, \ldots, v_n]\) be the standard \(n\)-simplex. A singular \(n\)-simplex in \(X\) is a continuous map \(\sigma\colon \Delta^n \to X\). The singular chain group \(C_n(X)\) is the free abelian group on the singular \(n\)-simplices in \(X\), and \(C_n(X)\coloneqq 0\) for \(n<0\). The boundary map \(\del_n\colon C_n(X)\to C_{n-1}(X)\) is the homomorphism with \[ \del_n \sigma \coloneqq \sum_{i=0}^n (-1)^i \ro{\sigma}{[v_0,\ldots, \hat v_i, \ldots, v_n]} \] on singular simplices, each face \([v_0,\ldots, \hat v_i, \ldots, v_n]\) being identified with \(\Delta^{n-1}\) by the order-preserving linear homeomorphism. The singular homology of \(X\) is \[ H_n(X) \coloneqq \ker \del_n / \im \del_{n+1} .\]

Universal cover (Tag D-BX3WD)

Let \(X\) be a topological space. A universal cover of \(X\) is a covering space \(p\colon \tilde X\to X\) with \(\tilde X\) simply connected [@Hat02].

Weak homotopy equivalence (Tag D-QMJHY)

A continuous map \(f\colon X\to Y\) of topological spaces is a weak homotopy equivalence if \(f_*\colon\pi_0(X)\to\pi_0(Y)\) is a bijection and, for every \(x_0\in X\) and every \(n\geq1\), the induced map on homotopy groups \[ f_*\colon\pi_n(X,x_0)\to\pi_n(Y,f(x_0)) \] is an isomorphism [@Hat02].

Weak topology determined by a family of subspaces (Tag D-2KICH)

Let \(X\) be a topological space and let \((X_\alpha)_{\alpha}\) be a family of subspaces with \(X = \Union_\alpha X_\alpha\). The weak topology determined by \((X_\alpha)_\alpha\) is the topology on \(X\) in which a subset \(A\subseteq X\) is closed if and only if \(A\intersect X_\alpha\) is closed in \(X_\alpha\) for every \(\alpha\).

Wedge sum (Tag D-IGUUS)

Let \((X_\alpha, x_\alpha)_{\alpha\in A}\) be a family of based topological spaces. The wedge sum \(\bigvee_{\alpha\in A} X_\alpha\) is the quotient space of the disjoint union \(\coprod_{\alpha\in A} X_\alpha\) obtained by identifying all the basepoints \(x_\alpha\) to a single point, which is its basepoint. For two based spaces, \(X\vee Y\) is the quotient of \(X\sqcup Y\) identifying \(x_0\) with \(y_0\).

Cone on a space (Tag D-II4M4)

Let \(X\) be a topological space and \(I = [0,1]\). The cone on \(X\) is the quotient space \[ CX\coloneqq(X\times I)/(X\times\ts{0}), \] in which \(X\times\ts{0}\) is collapsed to a point.

Suspension and reduced suspension (Tag D-RZKQD)

Let \(X\) be a topological space and \(I=[0,1]\). The suspension \(SX\) is the quotient of \(X\times I\) obtained by collapsing \(X\times\ts{0}\) to one point and \(X\times\ts{1}\) to another point [@Hat02]. For a based space \((X,x_0)\), the reduced suspension is \[ \Sigma X\coloneqq\frac{X\times I}{(X\times\ts{0})\cup(X\times\ts{1})\cup(\ts{x_0}\times I)}, \] the quotient of \(SX\) obtained by collapsing the segment \(\ts{x_0}\times I\) to a point [@Hat02].

Smash product (Tag D-R4ZCL)

Let \((X,x_0)\) and \((Y,y_0)\) be based spaces. Identify the wedge sum \(X\vee Y\) with the subspace \(X\times\ts{y_0}\cup\ts{x_0}\times Y\) of \(X\times Y\). The smash product is the quotient space \[ X\wedge Y\coloneqq(X\times Y)/(X\vee Y), \] based at the image of \(X\vee Y\) [@Hat02].

Moore space (Tag D-KC4BS)

Let \(G\) be an abelian group and \(n\geq 1\). A Moore space \(M(G, n)\) is a CW complex \(X\) with \[ \tilde H_i(X;\ZZ)\cong \begin{cases} G & i = n, \\ 0 & i\neq n. \end{cases} \]

Mapping cone (Tag D-TFVPD)

Let \(f\colon X\to Y\) be a continuous map with mapping cylinder \(M_f=\qty{(X\times I)\disjoint Y}/\qty{(x,1)\sim f(x)}\). The mapping cone of \(f\) is \[ C_f\coloneqq M_f/(X\times\ts{0})=\qty{CX\disjoint Y}/\qty{(x,1)\sim f(x)\text{ for }x\in X}, \] where \(CX=(X\times I)/(X\times\ts{0})\) is the cone on \(X\); that is, \(C_f\) is \(Y\) with \(CX\) attached along \(f\) [@Hat02].

Mapping cylinder (Tag D-RMQ7W)

Let \(f\colon X\to Y\) be a continuous map and \(I=[0,1]\). The mapping cylinder of \(f\) is the quotient space \[ M_f\coloneqq\qty{(X\times I)\disjoint Y}/\qty{(x,1)\sim f(x)\text{ for } x\in X} \] [@Hat02].

Mapping path space (Tag D-TPTOG)

Let \(f\colon X\to Y\) be a continuous map, \(I=[0,1]\), and \(Y^I\) the space of paths \(I\to Y\) with the compact-open topology. The mapping path space of \(f\) is the subspace \[ E_f\coloneqq\ts{(x,\gamma)\in X\times Y^I \st \gamma(0)=f(x)} \] of \(X\times Y^I\) [@Hat02].

Loop space (Tag D-T5Q3V)

Let \((X,x_0)\) be a based space and \(s_0\in S^1\) a basepoint. The loop space of \(X\) is the set \[ \Omega X\coloneqq\ts{\gamma\colon S^1\to X \st \gamma\text{ continuous},\ \gamma(s_0)=x_0} \] with the compact-open topology, based at the constant loop at \(x_0\) [@Hat02].

Eilenberg--MacLane space (Tag D-MENR4)

Let \(n\geq 1\) and let \(G\) be a group, abelian if \(n\geq 2\). An Eilenberg–MacLane space \(K(G, n)\) is a path connected CW complex \(X\) with homotopy groups \[ \pi_i(X)\cong \begin{cases} G & i = n, \\ 0 & i\neq n. \end{cases} \]

Tor functors (Tag D-4VGLT)

Let \(R\) be a commutative ring, let \(B\) be an \(R\)-module, and let \(n\geq 0\). The functor \(\Tor_n^R(\wait, B)\) is the \(n\)th left derived functor of \(\wait\tensor_R B\): \[ \Tor_n^R(A, B) \coloneqq L_n(\wait\tensor_R B)(A) = H_n(P_\bullet\tensor_R B) ,\] where \(P_\bullet \to A\) is a projective resolution of the \(R\)-module \(A\).

Inverse limit (Tag D-OKSJJ)

Let \((I,\leq)\) be a directed set. An inverse system of abelian groups indexed by \(I\) consists of abelian groups \(A_\alpha\) for \(\alpha\in I\) and homomorphisms \(f_{\beta\alpha}\colon A_\beta\to A_\alpha\) for \(\alpha\leq\beta\), with \(f_{\alpha\alpha}=\id_{A_\alpha}\) and \(f_{\beta\alpha}\circ f_{\gamma\beta}=f_{\gamma\alpha}\) for \(\alpha\leq\beta\leq\gamma\). Its inverse limit is the subgroup of compatible families \[ \varprojlim_\alpha A_\alpha\coloneqq\ts{(a_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}A_\alpha \st f_{\beta\alpha}(a_\beta)=a_\alpha\text{ whenever }\alpha\leq\beta}, \] with the projections \(\pi_\alpha\colon\varprojlim A_\alpha\to A_\alpha\).

Generating set of a module (Tag D-DUCA5)

Let \(R\) be a ring and \(M\) a left $R$-module. A subset \(S\subseteq M\) is a generating set for \(M\) if every \(x\in M\) is a finite \(R\)-linear combination of elements of \(S\): there exist \(n\geq 0\), \(s_1,\ldots,s_n\in S\), and \(r_1,\ldots,r_n\in R\) with \(x = \sum_{i=1}^n r_i s_i\).

Acyclic spaces and chain complexes (Tag D-3VEC5)

A topological space \(X\) is acyclic if its reduced singular homology vanishes, \(\tilde H_n(X;\ZZ) = 0\) for all \(n\). A chain complex \((C_*, \del)\) is acyclic if \(H_n(C_*) = 0\) for all \(n\).

Chain homotopy (Tag D-5KMYI)

Let \(R\) be a ring, let \((C_*, \del^C)\) and \((D_*, \del^D)\) be chain complexes of \(R\)-modules, and let \(f, g\colon C_*\to D_*\) be chain maps. A chain homotopy from \(f\) to \(g\) is a family of \(R\)-linear maps \(h_i\colon C_i\to D_{i+1}\), \(i\in\ZZ\), such that \[ f_i - g_i = \del^D_{i+1}\circ h_i + h_{i-1}\circ \del^C_i \quad\text{for all } i\in\ZZ .\] The chain maps \(f\) and \(g\) are chain homotopic if a chain homotopy from \(f\) to \(g\) exists.

Chain map (Tag D-36ONS)

Let \(R\) be a ring and let \((C_*, \del^C)\) and \((D_*, \del^D)\) be chain complexes of \(R\)-modules, with \(\del^C_i\colon C_i\to C_{i-1}\) and \(\del^D_i\colon D_i\to D_{i-1}\). A chain map \(f\colon C_*\to D_*\) is a family of \(R\)-linear maps \(f_i\colon C_i\to D_i\), \(i\in\ZZ\), such that \[ f_{i-1}\circ\del^C_i = \del^D_i \circ f_i \quad\text{for all } i\in\ZZ .\]

Coboundary (Tag D-D2K6Z)

Let \(X\) be a topological space and \(G\) an abelian group, and let \(C^n(X;G)\coloneqq\Hom(C_n(X), G)\) be the group of \(n\)-cochains, where \(C_n(X)\) is the group of singular $n$-chains with boundary map \(\del_n\colon C_n(X)\to C_{n-1}(X)\). The coboundary map \(\delta^n\colon C^n(X;G)\to C^{n+1}(X;G)\) is \(\delta^n(\varphi)\coloneqq\varphi\circ\del_{n+1}\). An \(n\)-cochain \(\varphi\) is a coboundary if \(\varphi\in B^n(X;G)\coloneqq\im\delta^{n-1}\), with \(B^0(X;G)\coloneqq 0\).

Cochain (Tag D-EILQL)

Let \(X\) be a topological space, \(R\) a commutative ring, \(p\geq 0\), and \(C_p(X;R)\) the free \(R\)-module on the singular \(p\)-simplices \(\sigma\colon\Delta^p\to X\). A \(p\)-cochain on \(X\) with coefficients in \(R\) is an \(R\)-linear map \(c\colon C_p(X;R)\to R\). The \(p\)-cochains form the \(R\)-module \(C^p(X;R)\coloneqq\Hom_R(C_p(X;R), R)\).

Cocycle (Tag D-MQTEG)

Let \(X\) be a topological space and \(G\) an abelian group. Let \(C^n(X;G)\coloneqq\Hom(C_n(X),G)\) be the group of singular $n$-cochains, where \(C_n(X)\) is the group of singular $n$-chains with boundary map \(\del_{n+1}\colon C_{n+1}(X)\to C_n(X)\), and let \(\delta^n\colon C^n(X;G)\to C^{n+1}(X;G)\) be the coboundary map \(\delta^n\psi\coloneqq\psi\circ\del_{n+1}\). An \(n\)-cochain \(\psi\in C^n(X;G)\) is an \(n\)-cocycle if \(\delta^n\psi=0\) [@Hat02]. The \(n\)-cocycles form the subgroup \(Z^n(X;G)\coloneqq\ker\delta^n\).

Cycle (Tag D-OH4ON)

Let \(X\) be a topological space, let \(C_n(X)\) be the group of singular $n$-chains of \(X\), and let \(\del_n\colon C_n(X)\to C_{n-1}(X)\) be the boundary map. An \(n\)-chain \(\alpha\in C_n(X)\) is an \(n\)-cycle if \(\del_n\alpha=0\) [@Hat02]. The \(n\)-cycles form the subgroup \(Z_n(X)\coloneqq\ker\del_n\).

Deck transformation (Tag D-4VGAW)

Let \(p\colon \tilde X\to X\) be a covering space. A deck transformation of \(p\) is a homeomorphism \(f\colon\tilde X\to\tilde X\) with \(p\circ f = p\). The deck transformations of \(p\) form a group \(\Deck(p)\) under composition.

Left derived functor (Tag D-6B77N)

Let \(R\) be a ring and let \(T\) be an additive covariant functor from \(R\)-modules to abelian groups. For an \(R\)-module \(A\), choose a projective resolution \(\cdots\to P_1\to P_0\to A\to 0\) and let \(P_\bullet\) be the complex \(\cdots \to P_1\to P_0\to 0\) with \(A\) deleted. The left derived functors of \(T\) are \[ L_n T(A) \coloneqq H_n\qty{T(P_\bullet)}, \qquad n\geq 0 .\]

Direct limit (Tag D-NODFN)

Let \((I,\leq)\) be a directed set. A directed system of abelian groups indexed by \(I\) consists of abelian groups \(A_\alpha\) for \(\alpha\in I\) and homomorphisms \(f_{\alpha\beta}\colon A_\alpha\to A_\beta\) for \(\alpha\leq\beta\), with \(f_{\alpha\alpha}=\id_{A_\alpha}\) and \(f_{\beta\gamma}\circ f_{\alpha\beta}=f_{\alpha\gamma}\) for \(\alpha\leq\beta\leq\gamma\). Its direct limit is the set \[ \varinjlim_\alpha A_\alpha\coloneqq\qty{\Disjoint_{\alpha\in I}A_\alpha}/\sim, \] where \(\sim\) is the equivalence relation generated by \(a\sim f_{\alpha\beta}(a)\) for \(a\in A_\alpha\) and \(\alpha\leq\beta\), with the group operation \([a]+[b]\coloneqq[f_{\alpha\gamma}(a)+f_{\beta\gamma}(b)]\) for \(a\in A_\alpha\), \(b\in A_\beta\), and any \(\gamma\geq\alpha,\beta\) [@Hat02]. For each \(\alpha\), the map \(\iota_\alpha\colon A_\alpha\to\varinjlim A_\alpha\), \(a\mapsto[a]\), is a homomorphism with \(\iota_\beta\circ f_{\alpha\beta}=\iota_\alpha\).

Direct product of modules (Tag D-XSBR2)

Let \(R\) be a ring and \((M_\alpha)_{\alpha\in A}\) a family of left \(R\)-modules. The direct product of the family is the set \[ \prod_{\alpha\in A} M_\alpha \coloneqq \ts{ (m_\alpha)_{\alpha\in A} \st m_\alpha \in M_\alpha \text{ for all } \alpha\in A } \] with componentwise addition and scalar multiplication, together with the projections \(\pi_\beta\colon\prod_{\alpha\in A} M_\alpha\to M_\beta\), \((m_\alpha)_{\alpha\in A}\mapsto m_\beta\), for \(\beta\in A\).

Direct sum of modules (Tag D-TZSG2)

Let \(R\) be a ring and \((M_\alpha)_{\alpha\in A}\) a family of \(R\)-modules with direct product \(\prod_{\alpha}M_\alpha\). The direct sum of the family is the submodule \[ \bigoplus_{\alpha\in A}M_\alpha\coloneqq\ts{(m_\alpha)_{\alpha\in A}\in\prod_{\alpha\in A}M_\alpha \st m_\alpha=0\text{ for all but finitely many }\alpha}, \] with the inclusions \(\iota_\beta\colon M_\beta\injects\bigoplus_\alpha M_\alpha\) sending \(m\) to the family with \(m\) in position \(\beta\) and \(0\) elsewhere [@DF04].

Exact sequence (Tag D-STPAM)

Let \(R\) be a ring and let \[ \cdots\to A_{n+1}\mapsvia{d_{n+1}}A_n\mapsvia{d_n}A_{n-1}\to\cdots \] be a sequence of \(R\)-modules and \(R\)-linear maps. The sequence is exact at \(A_n\) if \(\im d_{n+1}=\ker d_n\), and exact if it is exact at every term that has both an incoming and an outgoing map [@Hat02].

Ext group (Tag D-BYIZA)

Let \(A\) and \(G\) be abelian groups, and let \[ \cdots \to F_2 \to F_1 \to F_0 \to A \to 0 \] be a free resolution of \(A\). Applying \(\Hom(\wait, G)\) to \(\cdots\to F_1\to F_0\) gives a cochain complex \(0\to\Hom(F_0,G)\to\Hom(F_1,G)\to\cdots\), and the Ext group \(\Ext^n(A, G)\) is its \(n\)th cohomology group; it is independent of the choice of free resolution [@Hat02].

Flat module (Tag D-FS52P)

Let \(R\) be a commutative ring. An \(R\)-module \(M\) is flat if the functor \(M\otimes_R -\) from \(R\)-modules to \(R\)-modules is exact.

Free module (Tag D-LIEMF)

Let \(R\) be a ring. An $R$-module \(M\) is free if it has a basis: a family \((\beta_i)_{i\in I}\) in \(M\) such that every \(m\in M\) can be written as \(m=\sum_{i\in I} r_i\beta_i\) with \(r_i\in R\), all but finitely many zero, and \(\sum_{i\in I} r_i\beta_i = 0\) implies \(r_i = 0\) for all \(i\).

Group ring (Tag D-FHUV5)

Let \(R\) be a commutative ring and \(G = \ts{g_1, \ldots, g_n}\) a finite group. The group ring \(RG\) is the free \(R\)-module with basis the elements of \(G\), \[ RG\coloneqq\ts{ \sum_{i=1}^n a_i g_i \st a_i\in R }, \] with the multiplication that extends the group law \(R\)-bilinearly: \((ag)(bh)\coloneqq(ab)(gh)\) for \(a, b\in R\) and \(g, h\in G\).

Homologous cycles (Tag D-7UFN3)

Let \(X\) be a topological space and \(n\geq 0\). Two cycles \(\alpha, \beta \in Z_n(X)\) are homologous if there exists \(\gamma\in C_{n+1}(X)\) with \(\alpha - \beta = \del_{n+1}\gamma\).

Limit of a diagram (Tag D-QXER7)

Let \(J\) be a small category, \(\mathcal C\) a category, and \(F\colon J\to\mathcal C\) a functor. A cone over \(F\) is an object \(Y\) of \(\mathcal C\) with morphisms \(\psi_j\colon Y\to F(j)\) for \(j\in J\) such that \(F(u)\circ\psi_j=\psi_k\) for every morphism \(u\colon j\to k\) in \(J\). A limit of \(F\) is a cone \((L,(\pi_j)_{j\in J})\) over \(F\) such that for every cone \((Y,(\psi_j)_{j\in J})\) over \(F\) there is a unique morphism \(h\colon Y\to L\) with \(\pi_j\circ h=\psi_j\) for all \(j\in J\).

Linearly independent subset of a module (Tag D-5RWYR)

Let \(R\) be a ring and \(M\) an \(R\)-module. A subset \(S\subseteq M\) is linearly independent if for all \(n\geq 1\), all pairwise distinct \(s_1, \ldots, s_n\in S\) and all \(r_1, \ldots, r_n\in R\), \[ r_1 s_1 + \cdots + r_n s_n = 0 \implies r_1 = \cdots = r_n = 0 .\]

Long exact sequence (Tag D-UEWPN)

A long exact sequence is an exact sequence with infinitely many terms, indexed by \(\ZZ\) or by an infinite interval of \(\ZZ\).

Perfect pairing (Tag D-EGHL6)

Let \(R\) be a commutative ring and \(M\), \(N\), \(L\) \(R\)-modules. A pairing of \(M\) and \(N\) with values in \(L\) is an \(R\)-bilinear map \(p\colon M\times N\to L\), equivalently an \(R\)-linear map \(M\otimes_R N\to L\) on the tensor product. The pairing \(p\) is perfect if the \(R\)-linear map \[ \phi\colon M\to\Hom_R(N, L),\qquad \phi(m)(n)\coloneqq p(m, n), \] is an isomorphism.

Pullback of spaces (Tag D-VWYRN)

Let \(f\colon X\to Z\) and \(g\colon Y\to Z\) be continuous maps of topological spaces. The pullback of \(f\) and \(g\) is the subspace \[ X \cross_Z Y \coloneqq \ts{ (x,y) \in X\cross Y \st f(x) = g(y) } \] of the product space \(X\cross Y\), together with the restrictions \(p_X\colon X\cross_Z Y\to X\) and \(p_Y\colon X\cross_Z Y\to Y\) of the two projections.

Pushout of topological spaces (Tag D-5S7PK)

Let \(f\colon Z\to X\) and \(g\colon Z\to Y\) be continuous maps. The pushout of \(X \xleftarrow{f} Z \mapsvia{g} Y\) is the quotient space \[ X \disjoint_Z Y \coloneqq \qty{X \disjoint Y}/\sim ,\] where \(\sim\) is the equivalence relation generated by \(f(z)\sim g(z)\) for \(z\in Z\), together with the maps \(j_X\colon X\to X\disjoint_Z Y\) and \(j_Y\colon Y\to X\disjoint_Z Y\) induced by the inclusions.

Quasi-isomorphism (Tag D-HWN6T)

Let \(R\) be a ring and \(C_\bullet\), \(D_\bullet\) chain complexes of \(R\)-modules. A chain map \(f\colon C_\bullet\to D_\bullet\) is a quasi-isomorphism if the induced map \(H_n(f)\colon H_n(C_\bullet)\to H_n(D_\bullet)\) is an isomorphism for every \(n\in\ZZ\).

Projective resolution (Tag D-PKIY7)

Let \(R\) be a ring and \(M\) an \(R\)-module. A projective resolution of \(M\) is an exact sequence of \(R\)-modules \[ \cdots\to P_2\mapsvia{d_2}P_1\mapsvia{d_1}P_0\mapsvia{\varepsilon}M\to0 \] in which every \(P_i\) is a projective module. It is a free resolution if every \(P_i\) is a free module.