Homology theory and computations

Basic properties

$H_0(X)$ is a free abelian group on the set of path components of $X$. In particular, $H_0(X) \cong \ZZ$ if $X$ is path connected, and $H_0(X) \cong \ZZ^{\abs{\pi_0(X)}}$ if $X$ has finitely many path components.

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.

(Wedges of spheres)

For $n\geq 1$ and $k\geq 1$, \[ H_{n}\qty{\bigvee^{k} S^n} \cong \ZZ^k. \]

Give the finite wedge one $0$-cell and $k$ cells in dimension $n$. Its cellular chain group in degree $n$ is therefore $\ZZ^k$, with zero incoming and outgoing cellular differential in that degree, so \[ H_n\qty{\bigvee_{j=1}^k S^n}\cong \ZZ^k. \]

Homology does not commute with products: $H_2(S^1\cross S^1)\cong\ZZ$, while $H_2(S^1)\cross H_2(S^1)=0$. For a finite product, the Künneth theorem computes $H_k$. If the homology groups of $A$ are free, then \[ H_{k} (A\cross B) \cong \bigoplus_{i+j=k} H_{i}(A) \tensor H_{j}(B) \] and if the homology groups of $X_1,\ldots,X_{k-1}$ are free, iterating gives \[ H_{n}\qty{\prod_{j=1}^k X_{j}} \cong \bigoplus_{\mathbf{x} \in \mathcal{P}(n,k)} \bigotimes_{i=1}^{k} H_{x_{i}}(X_{i}), \] where $\mathcal P(n,k)$ is the set of $k$-tuples $\mathbf x=(x_1,\ldots,x_k)$ of nonnegative integers with $x_1+\cdots+x_k=n$.

Künneth isomorphism with free homology (Tag FF-UBJ3S)

With one factor homologically free over a PID, product homology is the graded tensor product.Suggest a slogan

Let $X, Y$ be CW complexes, let $R$ be a principal ideal domain, and suppose $H_i(X;R)$ is a free $R$-module for every $i$ (for example, $R$ a field). Then for each $k$ the cross product gives an isomorphism \[ \bigoplus_{i+j=k}H_{i}(X;R)\tensor_R H_{j}(Y;R) \xrightarrow{\ \sim\ } H_{k}(X\times Y;R) \] [@Hat02].

State the Kunneth isomorphism in the torsion-free case.

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].

(Cellular-chain quick checks)

If a CW complex has no $n\dash$cells, then its cellular chain group $C_n(X)$ is zero, hence $H_n(X)=0$.
If a CW complex has a single $0\dash$cell, then its cellular differential $d_1:C_1(X)\to C_0(X)$ is zero.

Computed homology groups

(Spheres)

\[ H_{i}(S^n) = \begin{cases} \ZZ & i = 0, n \\ 0 & \text{else}. \end{cases} \]

Real projective spaces

Homology of $\RP^2$ (Tag FF-SYCKI)

$\RP^2$ has $\ZZ$ in degree $0$, $\ZZ/2$ in degree $1$, and zero otherwise.Suggest a slogan

The integral homology groups of $\RP^2$ are \[ H_k(\RP^2;\ZZ) \cong \begin{cases} \ZZ & k = 0,\\ \ZZ/2 & k = 1,\\ 0 & \text{otherwise} \end{cases} \] [@Hat02].

What is $H_* \mathbb{RP}^2$, degree by degree?

Homology of $\RP^3$ (Tag FF-MJEUU)

$\RP^3$ has $\ZZ$ in degrees $0$ and $3$, $\ZZ/2$ in degree $1$, and zero otherwise.Suggest a slogan

The integral homology groups of $\RP^3$ are \[ H_k(\RP^3;\ZZ) \cong \begin{cases} \ZZ & k = 0, 3,\\ \ZZ/2 & k = 1,\\ 0 & \text{otherwise} \end{cases} \] [@Hat02].

What is $H_* \mathbb{RP}^3$?

Homology of $\RP^4$ (Tag FF-7LLAF)

$\RP^4$ has $\ZZ$ in degree $0$, $\ZZ/2$ in degrees $1$ and $3$, and zero otherwise.Suggest a slogan

The integral homology groups of $\RP^4$ are \[ H_k(\RP^4;\ZZ) \cong \begin{cases} \ZZ & k = 0,\\ \ZZ/2 & k = 1, 3,\\ 0 & \text{otherwise} \end{cases} \] [@Hat02].

What is $H_* \mathbb{RP}^4$?

Complex projective spaces

Homology of $\CP^2$ (Tag FF-NEJ3S)

$\CP^2$ has integral homology $\ZZ$ in degrees $0$, $2$, and $4$, and zero otherwise.Suggest a slogan

The integral homology groups of $\CP^2$ are \[ H_k(\CP^2;\ZZ) \cong \begin{cases} \ZZ & k = 0, 2, 4,\\ 0 & \text{otherwise} \end{cases} \] [@Hat02].

What is $H_* \mathbb{CP}^2$?

Surfaces

Homology of the Klein bottle (Tag FF-QOQ2K)

The Klein bottle has $H_0=\ZZ$, $H_1=\ZZ\oplus\ZZ/2$, and no higher integral homology.Suggest a slogan

The integral homology groups of the Klein bottle $K$ are \[ H_k(K;\ZZ) \cong \begin{cases} \ZZ & k = 0,\\ \ZZ\oplus\ZZ/2 & k = 1,\\ 0 & \text{otherwise} \end{cases} \] [@Hat02].

What is the homology of the Klein bottle, degree by degree?

Mayer-Vietoris

(Splitting over a free quotient)

Since $\ZZ^m$ is free, every exact sequence of abelian groups $0 \to \ZZ^n \to A \to \ZZ^m \to 0$ splits, so $A\cong \ZZ^{n}\oplus \ZZ^m$.

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 $X = A \cup B$.

(Homology of a connected sum)

For connected closed $n$-manifolds $M$ and $N$, $M\# N = (M\sm D)\union_{S^{n-1}} (N\sm D')$ for open $n$-balls $D\subseteq M$ and $D'\subseteq N$. The Mayer–Vietoris sequence for open neighborhoods of the two pieces, which deformation retract onto them and meet in a neighborhood of $S^{n-1}$ deformation retracting onto $S^{n-1}$, computes $H_*(M\# N)$ from $H_*(M\sm D)$, $H_*(N\sm D')$, and $H_*(S^{n-1})$.

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 $H^0(S^{n-1})\cong\ZZ$.

Let $n\geq 1$ and write $S^n = A \cup B$ with $A$ and $B$ open neighborhoods of the closed northern and southern hemispheres, chosen so that $A$ and $B$ are contractible and $A \cap B$ deformation retracts onto the equator $S^{n-1}$. The Mayer–Vietoris sequence in reduced cohomology contains

\[ \tilde H^{i-1}(A) \oplus \tilde H^{i-1}(B) \to \tilde H^{i-1}(S^{n-1}) \xrightarrow{\delta} \tilde H^i(S^n) \to \tilde H^i(A) \oplus \tilde H^i(B) .\]

Since $A$ and $B$ are contractible, their reduced cohomology vanishes, so

\[ 0 \to \tilde H^{i-1}(S^{n-1}) \xrightarrow{\delta} \tilde H^i(S^n) \to 0 \] is exact, and $\delta$ is an isomorphism $\tilde H^{i-1}(S^{n-1})\cong \tilde H^i(S^n)$.

Further exact sequences

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].

Relative homology

(Relative and cellular homology)

If $(X,A)$ is a good pair, then the quotient map induces \[ H_n(X,A)\cong \widetilde H_n(X/A). \]
The long exact sequence of a pair is \[ \cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\to H_{n-1}(A)\to\cdots. \]
For a CW complex $X$, the cellular filtration satisfies \[ H_j(X^{(k)},X^{(k-1)}) \cong \begin{cases} \ZZ[\theset{\text{$k$-cells of $X$}}] & j=k,\\ 0 & j\neq k, \end{cases} \] since $X^{(k)}/X^{(k-1)}$ is a wedge of $k$-spheres, one for each $k$-cell.

Exercises

$S^2\times\RP^4$ is not homotopy equivalent to $S^4\times\RP^2$ (Tag P-S7WVQ)

Show that $S^2 \cross \RP^4 \not\homotopic S^4 \cross \RP^2$.

Solution

Take coefficients in $\FF_2$, apply Kunneth to get \[ H^*(X_1; \FF_2) = \Extalgebra_{\FF_2}{x_2} \tensor_{\FF_2} {\FF_2\adjoin{y_1} \over \gens{ y_1^5 }} \\ H^*(X_2; \FF_2) = \Extalgebra_{\FF_2}{w_4} \tensor_{\FF_2} {\FF_2\adjoin{z_1} \over \gens{ z_1^3 }} .\]

Now use that any homotopy equivalence induces a graded ring isomorphism $f$, where $f(y_1) = z_1$ in $H^1$, but these have different orders.

$S^n$ is not a strong deformation retract of $B^{n+1}$ (Tag E-AUAOC)

Show that $S^n$ is not a strong deformation retract of $\bB^{n+1}$.