Homology · Qual Corpus

Singular homology $H_n(X)$ is the homology of the chain complex of formal sums of maps $\Delta^n\to X$, and homotopic maps induce equal maps on homology. For a CW complex $X$, cellular homology is the homology of the chain complex with $n$th group $H_n(X^n,X^{n-1})$, free on the $n$-cells, whose differentials are given by the degrees of attaching maps composed with collapse maps $S^{n-1}\to S^{n-1}$; it is isomorphic to singular homology. A pair $(X,A)$ has a long exact sequence $\cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\to H_{n-1}(A)\to\cdots$. Excision: for $Z\subseteq A\subseteq X$ with $\overline Z$ contained in the interior of $A$, the inclusion induces isomorphisms $H_n(X\setminus Z,A\setminus Z)\cong H_n(X,A)$. If $X$ is the union of the interiors of $A$ and $B$, the Mayer–Vietoris sequence $\cdots\to H_n(A\cap B)\to H_n(A)\oplus H_n(B)\to H_n(X)\to H_{n-1}(A\cap B)\to\cdots$ is exact. For a wedge of spaces whose basepoints have neighbourhoods deformation retracting onto them, $\tilde H_n(\bigvee_\alpha X_\alpha)\cong\bigoplus_\alpha\tilde H_n(X_\alpha)$, and the Künneth formula computes $H_n(X\times Y)$ from the homology of $X$ and $Y$. These tools give $\tilde H_n(S^k)\cong\ZZ$ for $n=k$ and $\tilde H_n(S^k)=0$ for $n\ne k$.

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 \[ H_n(X) \coloneqq \ker \del_n / \im \del_{n+1} .\]

The quotient is defined because $\del_n \circ \del_{n+1} = 0$.

[@Hat02].

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.

See [@Hat02].

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 $Z_n(X)\coloneqq\ker\del_n$.

With $B_n(X)\coloneqq\im\del_{n+1}$ the group of $n$-boundaries, the relation $\del_n\circ\del_{n+1}=0$ gives $B_n(X)\subseteq Z_n(X)$, and the singular homology group is $H_n(X)=Z_n(X)/B_n(X)$.

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 $\alpha - \beta = \del_{n+1}\gamma$.

Two cycles are homologous if and only if they represent the same class in the singular homology group $H_n(X)$.

[@Hat02].

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 $\del_n \alpha \in C_{n-1}(A)$.

A chain $\alpha\in C_n(X)$ is a relative cycle if and only if its image in $C_n(X,A)$ is a cycle for the boundary map induced by $\del_n$.

[@Hat02].

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 $\gamma \in C_n(A)$.

A chain $\alpha\in C_n(X)$ is a relative boundary if and only if its image in $C_n(X,A) = C_n(X)/C_n(A)$ is a boundary for the induced boundary map. The relative homology group $H_n(X,A)$ is isomorphic to the group of relative cycles in $C_n(X)$ modulo the subgroup of relative boundaries.

[@Hat02].

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 $H_n(X\mid x)\coloneqq H_n(X\mid\ts{x})$.

[@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 $(C_*^{\cell}(X), d_*)$.

The group $C_n^{\cell}(X)$ is free abelian with a basis in bijection with the $n$-cells of $X$, and $d_{n-1}\circ d_n = 0$. By the cellular boundary formula, for $n\geq 2$ the coefficient of the $(n-1)$-cell $e^{n-1}_\beta$ in $d_n(e^n_\alpha)$ is the degree of the composite $S^{n-1}_\alpha \to X^{n-1}\to X^{n-1}/\qty{X^{n-1}\sm e^{n-1}_\beta} \cong S^{n-1}_\beta$ of the attaching map of $e^n_\alpha$ and the quotient map. There is an isomorphism $H_n^{\cell}(X)\cong H_n(X)$ for every $n$ [@Hat02].

[@Hat02].

Excision (Tag T-FBMYQ)

For $Z\subseteq A\subseteq X$ with $\cl_X(Z)\subseteq A^\circ$, removing $Z$ from $X$ and $A$ does not change $H_*(X, A)$.Suggest a slogan

Given subspaces $Z \subseteq A \subseteq X$ with $\cl_X(Z) \subseteq \interiorof{A}$, the inclusion of pairs induces isomorphisms \[ H_n(X\sm Z,\, A\sm Z) \mapsvia{\sim} H_n(X, A) \qquad \text{for all } n .\] Equivalently, for subspaces $A, B\subseteq X$ with $X = \interiorof{A} \union \interiorof{B}$, the inclusion $(B, A\intersect B)\injects (X, A)$ induces isomorphisms $H_n(B, A\intersect B)\mapsvia{\sim} H_n(X,A)$ for all $n$. The two forms are related by $B = X\sm Z$ [@Hat02].

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 $\ZZ$.

Let $0\to A_\bullet\mapsvia{i}B_\bullet\mapsvia{j}C_\bullet\to0$ be a short exact sequence of chain complexes of abelian groups. There are homomorphisms $\del\colon H_n(C_\bullet)\to H_{n-1}(A_\bullet)$, the connecting homomorphisms, such that \[ \cdots\to H_n(A_\bullet)\mapsvia{i_*}H_n(B_\bullet)\mapsvia{j_*}H_n(C_\bullet)\mapsvia{\del}H_{n-1}(A_\bullet)\to\cdots \] is exact [@Hat02].

For a topological space $X$ and a subspace $A$, the short exact sequence $0\to C_\bullet(A)\to C_\bullet(X)\to C_\bullet(X,A)\to0$ of singular chain complexes gives the long exact sequence of the pair \[ \cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\mapsvia{\del}H_{n-1}(A)\to\cdots\to H_0(X,A)\to0 \] [@Hat02].

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 \[ \cdots\to H_n(A\cap B)\xrightarrow{\Phi} H_n(A)\oplus H_n(B)\xrightarrow{\Psi} H_n(X)\xrightarrow{\del} H_{n-1}(A\cap B)\to\cdots\to H_0(X)\to 0. \]

Let $X$ be a topological space and $A, B\subseteq X$ subspaces with $X = A^\circ\cup B^\circ$. Then $\del\colon H_n(X)\to H_{n-1}(A\cap B)$ is well defined and the Mayer–Vietoris sequence of $(A, B)$ is exact [@Hat02].

Let $C_n(A+B)\subseteq C_n(X)$ be the subgroup of sums of an $n$-chain in $A$ and an $n$-chain in $B$. Since $X = A^\circ\cup B^\circ$, the inclusion $C_\bullet(A+B)\hookrightarrow C_\bullet(X)$ induces isomorphisms on homology. The Mayer–Vietoris sequence is the long exact homology sequence of the short exact sequence of chain complexes \[ 0\to C_n(A\cap B)\xrightarrow{x\mapsto(x, -x)} C_n(A)\oplus C_n(B)\xrightarrow{(x, y)\mapsto x+y} C_n(A+B)\to 0, \] and $\del$ is its connecting homomorphism; since $\del z = 0$, the chain $\del x = -\del y$ lies in $A\cap B$ and is a cycle there.

When $A\cap B\neq\emptyset$ there is an exact sequence of the same form in reduced homology, and there is an exact sequence for singular cohomology with the arrows reversed.

Write the Mayer-Vietoris long exact sequence for $X = A \cup B$.

Reduced homology of a wedge sum (Tag PR-B6BB2)

Reduced homology sends wedge sums of good pointed spaces to direct sums.Suggest a slogan

Let $(X_\alpha, x_\alpha)$ be pointed spaces such that each $(X_\alpha, \theset{x_\alpha})$ is a good pair. Then the inclusions $X_\alpha\injects\bigvee_\alpha X_\alpha$ induce an isomorphism \[ \bigoplus_\alpha \tilde H_n(X_\alpha) \xrightarrow{\ \sim\ } \tilde H_n\qty{\bigvee_\alpha X_\alpha} \] for every $n$ [@Hat02].

The wedge sum $\bigvee_\alpha X_\alpha$ is the coproduct in the category of pointed topological spaces.

Suspension isomorphism for the cohomology of spheres (Tag PR-6PENU)

Reduced cohomology of spheres shifts down by one under desuspension.Suggest a slogan

For $n\geq 1$ and all $i$, the reduced cohomology groups of spheres satisfy \[ \tilde H^i(S^n) \cong \tilde H^{i-1}(S^{n-1}) \] [@Hat02].

The statement fails for unreduced cohomology: for $n\geq 2$, $H^1(S^n) = 0$ while $H^0(S^{n-1})\cong\ZZ$.

Künneth formula for homology (Tag T-TZ3X7)

Product homology is tensor product plus a one-degree-shifted Tor correction.Suggest a slogan

Let $X, Y$ be CW complexes and $R$ a principal ideal domain. For each $k$ there is a natural short exact sequence \[ 0 \to \bigoplus_{i+j=k} H_{i}(X; R) \tensor_{R} H_{j}(Y; R) \to H_{k}(X\cross Y; R) \to \bigoplus_{i+j=k-1} \Tor^R_1\qty{H_{i}(X; R), H_{j}(Y; R)} \to 0 ,\] and it splits, though not naturally: \[ H_{k}(X\cross Y; R) \cong \qty{ \bigoplus_{i+j = k} H_{i}(X;R) \tensor_R H_{j}(Y;R) } \oplus \bigoplus_{i+j = k-1}\Tor^R_1\qty{H_{i}(X;R), H_{j}(Y;R)} \] [@Hat02].

Universal coefficient theorems (Tag T-2W5WN)

Changing coefficients adds Tor to homology and Ext to cohomology.Suggest a slogan

Let $X$ be a topological space and $G$ an abelian group. For each $i$ there are natural short exact sequences \[ 0 \to H_i(X;\ZZ)\tensor G \to H_i(X; G) \to \Tor(H_{i-1}(X;\ZZ), G) \to 0 \] [@Hat02] and \[ 0 \to \Ext(H_{i-1}(X;\ZZ), G) \to H^i(X;G) \to \Hom(H_i(X;\ZZ), G) \to 0 \] [@Hat02]. Both sequences split, though not naturally, so \[ H_i(X;G) \cong \qty{H_i(X;\ZZ)\tensor G} \oplus \Tor(H_{i-1}(X;\ZZ), G), \qquad H^i(X;G) \cong \Hom(H_i(X;\ZZ), G) \oplus \Ext(H_{i-1}(X;\ZZ), G) .\]

If $H_i(X;\ZZ)$ and $H_{i-1}(X;\ZZ)$ are finitely generated with torsion subgroups $T_i$ and $T_{i-1}$, then \[ H^i(X;\ZZ) \cong \qty{H_i(X;\ZZ)/T_i} \oplus T_{i-1} \] [@Hat02]. If $F$ is a field, then $H^i(X;F) \cong \Hom_F(H_i(X;F), F)$ [@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 \[ \tilde H_i(X;\ZZ)\cong \begin{cases} G & i = n, \\ 0 & i\neq n. \end{cases} \]

For every abelian group $G$ and $n\geq 1$ a Moore space $M(G, n)$ exists. Choose a free resolution $0\to F_1\mapsvia{\varphi} F_0\to G\to 0$ with a basis $(a_\alpha)_{\alpha}$ of the free abelian group $F_0$ and a basis $(b_\beta)_{\beta}$ of the free abelian group $F_1$; such a resolution exists because every subgroup of a free abelian group is free. Let $X^n\coloneqq\bigvee_\alpha S^n_\alpha$ be a wedge of $n$-spheres, one for each $a_\alpha$, and identify $H_n(X^n;\ZZ)$ with $F_0$ by sending the class of $S^n_\alpha$ to $a_\alpha$. For each $\beta$, attach an $(n+1)$-cell $e^{n+1}_\beta$ along a map $f_\beta\colon S^n\to X^n$ with $(f_\beta)_*[S^n] = \varphi(b_\beta)$, which exists because the Hurewicz map $\pi_n(X^n)\to H_n(X^n;\ZZ)$ is surjective; composed with the collapse $X^n\to S^n_\alpha$, the map $f_\beta$ has degree equal to the $a_\alpha$-coordinate of $\varphi(b_\beta)$. The resulting CW complex $X$ has cellular chain complex $0\to F_1\mapsvia{\varphi}F_0\to 0$ in degrees $n+1$ and $n$, so $\tilde H_n(X;\ZZ)\cong\coker\varphi\cong G$ and $\tilde H_{n+1}(X;\ZZ)\cong\ker\varphi = 0$ [@Hat02].

The sphere $S^n$ is an $M(\ZZ, n)$. For $m\geq 2$, the space $S^n\cup_m e^{n+1}$ obtained by attaching an $(n+1)$-cell to $S^n$ along a map $S^n\to S^n$ of degree $m$ is an $M(\ZZ/m, n)$.

See [@Hat02].

Hurewicz theorem (Tag T-GL7E6)

For a highly connected space, the first nonzero homotopy group agrees with the first nonzero homology group.Suggest a slogan

Let $n\geq 2$ and let $X$ be an $(n-1)$-connected space. Then $\tilde H_i(X) = 0$ for $i < n$, and the Hurewicz map $\pi_n(X)\to H_n(X)$ is an isomorphism [@Hat02]. For $n = 1$ and $X$ path-connected, the Hurewicz map $\pi_1(X)\to H_1(X)$ induces an isomorphism from the abelianization of $\pi_1(X)$ onto $H_1(X)$ [@Hat02].

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