Exact Sequences and Relative Homology

Exact Sequences

Let’s recall the definition of exact sequences in the context of abelian groups:

A sequence of morphisms between objects in an abelian category $\cdots \stackrel{f_{n-1}}{\to }{A}_n\stackrel{f_n}{\to }{A}_{n+1}\stackrel{f_{n+1}}{\to }\dots$ is exact at $A_n$ if $\text{im}(f_{n-1})=\ker (f_n)$. The sequence is an exact sequence if it is exact at every object.

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Short Exact Sequence

We call an exact sequence in form of $0\to A\stackrel{i}{\to }B\stackrel{j}{\to }C\to 0$ as short exact sequence, more specifically with $i$ injective and $j$ surjective.

Theorem

For a short exact sequence $0\to A\stackrel{i}{\to }B\stackrel{j}{\to }C\to 0$, The sequence of homology groups $\cdots \to H_n(A)\stackrel{i_{\ast }}{\to }{H}_n(B)\stackrel{j_{\ast }}{\to }{H}_n(C)\stackrel{\partial }{\to }{H}_{n-1}(A)\stackrel{i_{\ast }}{\to }{H}_{n-1}(B)\to \cdots$ is exact

Long Exact Sequence of a Good Pair

Good Pair

A pair $(X,A)$ with $A\subset X$ is called a good pair if $A$ is a non-empty closed subset of $X$ that has an open neighborhood which deformation retracts onto $A$.

Long Exact Sequence of a Good Pair

For a good pair $(X,A)$, there is a long exact sequence in homology: $\cdots \to H_n(A)\stackrel{i_{\ast }}{\to }{H}_n(X)\stackrel{j_{\ast }}{\to }{\tilde{H}}_n(X/A)\stackrel{\partial }{\to }{H}_{n-1}(A)\stackrel{i_{\ast }}{\to }\cdots$ The maps are:

• $i_{\ast }$: induced by the inclusion map $i:A\hookrightarrow X$.
• $j_{\ast }$: induced by the quotient map $j:X\to X/A$.
• $\partial$: the boundary map.

Optionally, we can write the sequence in reduced homology as: $\cdots \to {\tilde{H}}_n(A)\stackrel{i_{\ast }}{\to }{\tilde{H}}_n(X)\stackrel{j_{\ast }}{\to }{\tilde{H}}_n(X/A)\stackrel{\partial }{\to }{\tilde{H}}_{n-1}(A)\stackrel{i_{\ast }}{\to }\cdots$ But note that ${\tilde{H}}_n(X/A)$ is always reduced.

To prove this theorem we need the notion of relative homology.

Relative Chains and Homology

For any pair $(X,A)$, the relative chain groups are defined as the quotient $C_n(X,A):=C_n(X)/C_n(A)$. The boundary map on $C_n(X)$ induces a boundary map $d_n:C_n(X,A)\to C_{n-1}(X,A)$, turning it into a chain complex. The homology of this complex, denoted $H_n(X,A)$, is called the relative homology of the pair $(X,A)$.

The long exact sequence of a good pair can be derived from the following:

Theorem

For any pair $(X,A)$, there is a long exact sequence relating their homology groups: $\cdots \to H_n(A)\to H_n(X)\to H_n(X,A)\to H_{n-1}(A)\to \cdots .$

$(X,A,B),\text{where}B\subset A\subset X$ $\cdots \to H_n(A,B)\to H_n(X,B)\to H_n(X,A)\to H_{n-1}(A,B)\to \cdots$ is exact

The prove of this theorem directly follow prove of deriving long exact sequence from short exact sequence. We also have generalization for triple

Lemma

For a good pair $(X,A)$, the relative homology is isomorphic to the reduced homology of the quotient space: $H_n(X,A)\cong {\tilde{H}}_n(X/A).$

Excision Theorem

Given subspaces $Z\subset A\subset X$ such that the closure of $Z$ is contained in the interior of $A$, then the inclusion $(X-Z,A-Z)\to (X,A)$ induces isomorphism $H_n(X-Z,A-Z)\to H_n(X,A)$ for all $n$. Equivalently, for subspaces $A,B\subset X$ whose interiors cover $X$, the inclusion $(B,A\cap B)\to (X,A)$ induces isomorphisms $H_n(B,A\cap B)\to H_n(X,A)$ for all n.

Applications

Corollary

The reduced homology of the n-sphere $S^n$ is given by: ${\tilde{H}}_k(S^n)\cong \{\begin{array}{ll}\mathbb{Z}& k=n\\ 0& k\ne n\end{array}$

Proof We will prove by induction on $n$. Base Case (n=0): $S^0$ consists of two points, $\{-1,1\}$. $H_0(S^0)\cong \mathbb{Z}\oplus \mathbb{Z}$. The reduced homology is ${\tilde{H}}_0(S^0)\cong \mathbb{Z}$ and ${\tilde{H}}_k(S^0)=0$ for $k>0$. The formula holds. Inductive Step: Assume the formula holds for $n-1$. Consider the good pair $(D^n,S^{n-1})$. The quotient space $D^n/S^{n-1}$ is homeomorphic to $S^n$. The long exact sequence in reduced homology is: $\cdots \to {\tilde{H}}_k(D^n)\to {\tilde{H}}_k(S^n)\stackrel{\partial }{\to }{\tilde{H}}_{k-1}(S^{n-1})\to {\tilde{H}}_{k-1}(D^n)\to \cdots .$Since the disk $D^n$ is contractible, its reduced homology groups are all zero: ${\tilde{H}}_k(D^n)=0$ for all $k$. The sequence simplifies to: $\cdots \to 0\to {\tilde{H}}_k(S^n)\stackrel{\partial }{\to }{\tilde{H}}_{k-1}(S^{n-1})\to 0\to \cdots .$Exactness implies that the connecting homomorphism $\partial :{\tilde{H}}_k(S^n)\to {\tilde{H}}_{k-1}(S^{n-1})$ is an isomorphism for all $k$. By the inductive hypothesis, ${\tilde{H}}_{k-1}(S^{n-1})$ is $\mathbb{Z}$ for $k-1=n-1$ (i.e., $k=n$) and $0$ otherwise. Therefore, the same is true for ${\tilde{H}}_k(S^n)$, completing the induction. $\square$

The long exact sequence can be used to prove Brouwer’s fixed point theorem for higher dimensions:

Brouwer's Fixed Point Theorem

Any continuous map $f:D^n\to D^n$ has a fixed point.

Proof Assume there exists a map $f:D^n\to D^n$ with no fixed points. We can then define a retraction $r:D^n\to \partial D^n=S^{n-1}$ by sending a point $x$ to the intersection of the ray from $f(x)$ through $x$ with the boundary sphere $S^{n-1}$ (just like what we did here). Let $i:S^{n-1}\hookrightarrow D^n$ be the inclusion map. The composition $r\circ i:S^{n-1}\to S^{n-1}$ is the identity map. Note that $(S^n,S^{n-1})$ is a good pair, so ${\tilde{H}}_{n-1}(S^{n-1})\stackrel{i_{\ast }}{\to }{\tilde{H}}_{n-1}(D^n)\stackrel{r_{\ast }}{\to }{\tilde{H}}_{n-1}(S^{n-1})$is an exact sequence. However, ${\tilde{H}}_{n-1}(S^{n-1})\cong \mathbb{Z}$, ${\tilde{H}}_{n-1}(D^n)\cong 0$ and ${\tilde{H}}_{n-1}(S^{n-1})\cong \mathbb{Z}$, so it is impossible to compose to the identity, yielding a contradiction. $\square$

Corollary

For a wedge sum ${\bigvee }_{\alpha }{X}_{\alpha }$, the inclusions $i_{\alpha }:X_{\alpha }\hookrightarrow {\bigvee }_{\alpha }{X}_{\alpha }$ induce an isomorphism ${⨁}_{\alpha }{i}_{\alpha \ast }:{⨁}_{\alpha }{\tilde{H}}_n(X_{\alpha })\to {\tilde{H}}_n\left({\bigvee }_{\alpha }{X}_{\alpha }\right)$, provided that the wedge sum is formed at basepoints $x_{\alpha }\in X_{\alpha }$ such that the pairs $(X_{\alpha },x_{\alpha })$ are good.