This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
We now turn to tools for computing homology in practice. Computing the homology of an arbitrary space directly from the definition is essentially impossible, so we must develop methods for decomposing a large space into smaller pieces and recovering the homology of the whole from that of the parts. The most intuitive analogue is the Seifert–van Kampen theorem (§Covering Spaces, ⁋Theorem 13), where we observed that the functor \(\pi_1:\Top \rightarrow \Grp\) preserves colimits. Now the abelianization functor \(\ab:\Grp \rightarrow \Ab\) is the left adjoint of the forgetful functor \(U:\Ab \rightarrow \Grp\) ([Algebraic Structures] §Abelian Groups, ⁋Proposition 7), and left adjoints preserve colimits ([Category Theory] §Adjoint Functors, ⁋Theorem 9); hence the first homology functor \(H_1:\Top \rightarrow \Ab\), being their composite (§Covering Spaces, ⁋Theorem 15), must also preserve colimits. In particular, as in §Covering Spaces, ⁋Corollary 14, let a topological space \(X\) be the union of two connected open subsets \(U,V\). In the category \(\Ab\), the pushout of two abelian groups is the coequalizer of their direct sum, so the isomorphism
\[H_1(X)=H_1(U\cup V)\cong \frac{H_1(U)\oplus H_1(V)}{\left\langle (f(x),-g(x))\mid x\in H_1(U\cap V)\right\rangle}\tag{1}\]must hold. In this post we treat this in a more general setting.
Relative homology
To do so, we first need a generalization of homology itself. For a space \(X\) and an arbitrary subspace \(A\), define the \(k\)-th relative chain group \(C_k(X,A)\) as the quotient
\[C_k(X,A):=C_k(X)/C_k(A)\]It is then straightforward to verify that the boundary map \(\partial_k:C_k(X) \rightarrow C_{k-1}(X)\) induces a map \(C_k(X,A) \rightarrow C_{k-1}(X,A)\) between the quotient groups. This yields the chain complex
\[\cdots \longrightarrow C_k(X,A)\overset{\partial}{\longrightarrow} C_{k-1}(X,A)\longrightarrow\cdots\]Definition 1 The \(k\)-th homology \(H_k(X,A)\) of the above chain complex is called the relative homology.
Now consider the short exact sequence in \(\Ch_{\geq 0}(\Ab)\)
\[0 \rightarrow C_\bullet(A) \rightarrow C_\bullet(X) \rightarrow C_\bullet(X,A) \rightarrow 0\]By [Homological Algebra] §Long Exact Sequences, ⁋Theorem 1, we obtain the long exact sequence
\[\cdots \rightarrow H_k(A) \rightarrow H_k(X) \rightarrow H_k(X,A)\rightarrow H_{k-1}(A) \rightarrow \cdots\]Here the connecting map \(H_k(X,A) \rightarrow H_{k-1}(A)\) is nothing more than the boundary map applied to any cycle (or any representative thereof) in \(H_k(X,A)\). Moreover, if a continuous map \(f:X \rightarrow Y\) satisfies \(f(A)\subseteq B\), then \(f\) induces not only the chain map \(C_\bullet(X)\rightarrow C_\bullet(Y)\) but also \(C_\bullet(A) \rightarrow C_\bullet(B)\), and from the commutativity of the diagram
we obtain an induced chain map \(C_\bullet(X,A) \rightarrow C_\bullet(Y,B)\). Thus any \(f:(X,A) \rightarrow (Y,B)\) satisfying these conditions induces a map \(H_k(f):H_k(X,A) \rightarrow H_k(Y,B)\) in homology. Applying §Homotopy, ⁋Proposition 6 to \(X\) and \(A\) separately and using [Homological Algebra] §Diagram Chasing, ⁋Corollary 2, we see that homotopic continuous maps \(f,g\) satisfying this condition induce the same map in homology.
Excision theorem
Intuitively, the relative homology \(H_\bullet(X,A)\) of a pair \((X,A)\) is the homology of the chain complex \(C_\bullet(X,A)\), which is the cokernel of the inclusion \(C_\bullet(A)\hookrightarrow C_\bullet(X)\) in \(\Ch_{\geq 0}(\Ab)\). Since information about \(C_\bullet(A)\) is lost upon taking the quotient, one expects that removing a subset contained in \(A\) should not change the relative homology. This is indeed true under the following mild hypothesis.
Theorem 2 (Excision theorem) Let \(A\) be a subspace of a space \(X\), and let \(Z\) be a subspace of \(A\) satisfying \(\cl Z\subseteq \interior A\). Then the inclusion
\[(X\setminus Z, A\setminus Z)\hookrightarrow (X,A)\]induces an isomorphism
\[H_k(X\setminus Z, A\setminus Z)\rightarrow H_k(X,A)\]However, despite its intuitive plausibility, the proof of this theorem involves some technicalities, so we omit it here.
On the other hand, in geometric situations we already know a way to discard the information contained in \(A\): the quotient space \(X/A\) obtained by collapsing \(A\) to a point. It is therefore reasonable to conjecture a relation between the homology \(H_k(X/A)\) and the relative homology \(H_k(X,A)\). As with the theorem above, this requires that \(A\) not be too pathological.
Definition 3 For a space \(X\) and a subspace \(A\), the pair \((X,A)\) is called a good pair if \(A\) is closed and there exists an open set \(U\) in \(X\) such that \(A\subset U\) and \(A\) is a strong deformation retract of \(U\).
Let \((X,A)\) be a good pair, and let \(U\) be an open set satisfying the hypothesis of Definition 3. In the diagram
each row is exact and the first two columns are exact as well; hence by [Homological Algebra] §Diagram Chasing, ⁋Corollary 7 we obtain a short exact sequence of chain complexes
\[0\rightarrow C_\bullet(U,A)\rightarrow C_\bullet(X,A)\rightarrow C_\bullet(X,U)\rightarrow 0\]and the associated long exact sequence
\[\cdots \rightarrow H_k(U,A) \rightarrow H_k(X,A)\rightarrow H_k(X,U)\rightarrow H_{k-1}(U,A)\rightarrow \cdots\]Since \(A\) is a strong deformation retract of \(U\), we have \(H_k(U,A)=0\) for all \(k\), and therefore \(H_k(X,A)\cong H_k(X,U)\) for all \(k\).
On the other hand, for any closed subspace \(A\) the quotient space \(X/A\) obtained by collapsing \(A\) to a point is well defined, and the projection \(X \rightarrow X/A\) sends \(A\) to the single point \([A]\) and is a homeomorphism outside \(A\). Applying the same argument to the inclusions
\[\{[A]\}\subseteq U/A\subseteq X/A\]we see that since \(U/A\) strong deformation retracts onto the point \([A]\), the fact that \(H_k(U/A,[A])=0\) yields the isomorphism
\[H_k(X/A, [A])\cong H_k(X/A, U/A)\]which fits into the following diagram induced by the quotient map
Now, since \((X,A)\) is a good pair, the inclusions \(A\subset U\subset X\) satisfy the condition \(\cl A\subseteq \interior U\) of Theorem 2; hence the map \(H_k(X\setminus A, U\setminus A)\rightarrow H_k(X,U)\) induced by the inclusion
\[(X\setminus A, U\setminus A)\hookrightarrow (X,U)\]is an isomorphism. Similarly, applying Theorem 2 to the inclusions \(\{[A]\}\subseteq U/A\subseteq X/A\), we see that the map
\[H_k((X/A)\setminus [A], (U/A)\setminus [A])\]is an isomorphism. These also fit into the following diagram induced by the quotient map
where the left-hand vertical map \(H_k(X\setminus A, U\setminus A)\rightarrow H_k((X/A)\setminus [A], (U/A)\setminus [A])\) is an isomorphism because the quotient map \(p:X\rightarrow X/A\) is a homeomorphism outside \(A\). Combining these results, we obtain the isomorphism
\[H_k(X,A)\cong H_k(X/A,[A])\tag{2}\]On the other hand, by §Homology, ⁋Proposition 11 and [Homological Algebra] §Diagram Chasing, ⁋Corollary 2, for any space \(X\) and a point \(x\in X\) the long exact sequence
\[\begin{aligned}\cdots &\rightarrow H_k(x)\rightarrow H_k(X)\rightarrow H_k(X,x) \rightarrow H_{k-1}(x)\rightarrow\cdots \\\cdots&\rightarrow H_1(x)\rightarrow H_1(X) \rightarrow H_1(X,x) \rightarrow H_0(x) \rightarrow H_0(X)\rightarrow H_0(X,x)\rightarrow 0\end{aligned}\]yields an isomorphism \(H_k(X)\cong H_k(X,x)\) for all \(k>1\). Moreover, examining the long exact sequence for \(k=1\),
\[0 \rightarrow H_1(X) \rightarrow H_1(X, x) \overset{\partial}{\longrightarrow} H_{0}(x) \overset{\iota_\ast}{\longrightarrow} H_{0}(X)\]we see that \(\iota_\ast\) sends the generator of \(H_0(x)\) to the path component of \(X\) containing \(x\), hence is injective; therefore \(\partial\) is the zero map, and we obtain the same isomorphism \(H_1(X)\cong H_1(X,x)\).
From the injectivity of \(\iota_\ast\) we also obtain the short exact sequence
\[0 \rightarrow H_0(x)\rightarrow H_0(X) \rightarrow H_0(X,x)\rightarrow 0\]and hence the isomorphism \(H_0(X,x)\cong H_0(X)/\mathbb{Z}\). Geometrically, this amounts to removing the path component of \(H_0(X)\) that contains \(x\). For notational convenience, we define the reduced homology \(\widetilde{H}_k(X)\) for a fixed \(x\in X\) by
\[\widetilde{H}_k(X)=H_k(X,x)\]Then the right-hand side of isomorphism (2) can be replaced by \(\widetilde{H}_k(X/A)\), giving the following statement.
Proposition 4 For a good pair \((X,A)\), the quotient map \(X \rightarrow X/A\) induces an isomorphism
\[H_k(X,A)\cong \widetilde{H}_k(X/A)\]for every \(k\).
Simplicial homology and singular homology
Since relative homology was defined by taking the cokernel of the monomorphism \(C_\bullet(A)\rightarrow C_\bullet(X)\) in \(\Ch_{\geq 0}(\Ab)\), the same procedure can be carried out for \(C^\Delta_\bullet(A) \rightarrow C^\Delta_\bullet(X)\). The result is the simplicial-homology version of relative homology, denoted \(H_n^\Delta(X,A)\). Because simplicial homology uses “non-singular” chains, the inclusion
\[C_\bullet^\Delta(X) \rightarrow C_\bullet(X)\]induces a canonical homomorphism
\[H_\bullet^\Delta(X)\rightarrow H_\bullet(X)\tag{3}\]and similarly we have canonical homomorphisms
\[H_\bullet^\Delta(A)\rightarrow H_\bullet^\Delta(A),\qquad H_\bullet^\Delta(X,A)\rightarrow H_\bullet(X,A)\]By [Homological Algebra] §Long Exact Sequences, ⁋Proposition 2, these fit into the commutative diagram
Using this, we obtain the following.
Theorem 5 For any \(\Delta\)-complex \(X\), the homomorphism in (3) is an isomorphism.
Proof
We sketch the overall strategy. Using the commutative diagram above, consider the filtration
\[X_0\subset X_1\subset\cdots\subset X_l=X\]defined by the \(\Delta\)-complex structure on \(X\), and apply [Homological Algebra] §Diagram Chasing, ⁋Corollary 2 to the diagram
to run an induction. For the induction step, it suffices to show that for every \(n\) and every \(k\), the homomorphism between relative homologies
\[H_n^\Delta(X^k, X^{k-1})\rightarrow H_n(X^k, X^{k-1})\]is an isomorphism. Once this is assumed, §Homology, ⁋Proposition 10 gives \(H_n^\Delta(X^0)\cong H_n(X^0)\) for all \(n\) when \(k=1\); then by [Homological Algebra] §Diagram Chasing, ⁋Corollary 2 we obtain \(H_n^\Delta(X^1)\cong H_n(X^1)\), and inductively we construct the desired isomorphism for larger \(k\).
First, by definition \(C_\bullet^\Delta(X^k, X^{k-1})\) is nontrivial only when \(n=k\); hence \(H_n^\Delta(X^k, X^{k-1})\) is a nontrivial free abelian group (generated by the \(k\)-simplices) only when \(n=k\), and is trivial otherwise.
A similar result holds for singular homology: specifically, \(H_n(\Delta^k,\partial\Delta^k)\) is a free abelian group only when \(n=k\), and its generator is \(\id:\Delta^k \rightarrow \Delta^k\). To verify this, let \(\Lambda\) be obtained from \(\Delta^k\) by removing one of its \((k-1)\)-dimensional faces, and consider the long exact sequence for the triple \((\Delta^k, \partial\Delta^k, \Lambda)\)
\[\cdots\rightarrow H_n(\Delta^k,\Lambda)\rightarrow H_n(\Delta^k, \partial\Delta^k)\rightarrow H_{n-1}(\partial\Delta^k, \Lambda)\rightarrow H_{n-1}(\Delta^k,\Lambda)\rightarrow \cdots\]Since \(\Delta^k\) deformation retracts onto \(\Lambda\), we have \(H_\bullet(\Delta^k,\Lambda)=0\), and therefore \(H_k(\Delta^k, \partial\Delta^k)\cong H_{n-1}{\partial\Delta^k,\Lambda}\); moreover, for the good pair \((\partial\Delta^k,\Lambda)\) the quotient space \(\partial\Delta^k/\Lambda\) is homeomorphic to \(\Delta^{k-1}/\partial\Delta^{k-1}\). Using these facts we obtain
\[H_k(\Delta^k, \partial\Delta^{k})\cong H_{k-1}(\Delta^{k-1}, \partial\Delta^{k-1})\]and the desired result follows inductively.
Examining this process, we see that the generator of \(H_k(\Delta^k,\partial\Delta^k)\) in singular homology is exactly the \(k\)-simplex \(\Delta^k\) itself. Since the pair \((X^k,X^{k-1})\) is a union of such pairs \((\Delta^k,\partial\Delta^k)\), §Homology, ⁋Proposition 9 yields the desired conclusion.
Mayer–Vietoris sequence
Although we have omitted the proof for reasons of length, the excision theorem Theorem 2 is a powerful tool in homology theory. For instance, in the proof of Theorem 5 we used excision to ignore simplices of dimension less than \(k-1\), which made an inductive argument possible with the homology of a one-point space as the base case. This process essentially exhibits all the properties that homology ought to satisfy; axiomatizing them leads to the following.
Definition 6 (Eilenberg–Steenrod axioms) Let \(H_k\) be functors from the category of pairs of topological spaces to the category of abelian groups, and let
\[\partial:H_k(X,A)\rightarrow H_{k-1}(A,\emptyset):=H_{k-1}(A)\]be natural transformations. The Eilenberg–Steenrod axioms are the following.
- (Homotopy) If two maps \((X,A) \rightarrow (Y,B)\) are homotopic, then the two induced homomorphisms \(H_k(X,A) \rightarrow H_k(Y,B)\) coincide.
- (Excision) For \((X,A,Z)\) satisfying the conditions of Theorem 2, the inclusion \((X\setminus Z, A\setminus Z)\hookrightarrow (X,A)\) induces an isomorphism.
- (Dimension) For a one-point space \(\ast\), we have \(H_k(\ast)=0\) for all \(k>0\).
- (Additivity) If \(X=\coprod X_\alpha\), then \(H_k(X)\cong\bigoplus H_k(X_\alpha)\).
-
(Exactness) For each pair \((X,A)\), the two inclusions \((A,\emptyset) \hookrightarrow (X,\emptyset)\) and \((X,\emptyset)\hookrightarrow (X,A)\) fit into the long exact sequence
\[\cdots \rightarrow H_k(A)\rightarrow H_k(X) \rightarrow H_k(X,A) \rightarrow H_{k-1}(A)\rightarrow \cdots\]
Eilenberg and Steenrod proved that any homology theory defined in this way is naturally isomorphic to any other, provided the coefficient group \(H_0(\ast)\) is fixed. For example, we previously proved that simplicial and singular homology agree on \(\Delta\)-complexes; inspecting that proof step by step, one verifies that what we used were precisely the axioms of Definition 6. For practical computations it is convenient to introduce cellular homology on CW complexes; this homology also satisfies the above axioms, and therefore yields the same calculations as simplicial and singular homology.
All these homology theories have coefficient group \(\mathbb{Z}\); however, replacing \(\mathbb{Z}\) by an arbitrary abelian group \(A\) leaves every axiom in Definition 6 intact. Indeed, had we defined singular or simplicial homology using not free abelian groups
\[C^\Delta_\bullet(X),\qquad C_\bullet(X)\]but free \(A\)-modules
\[C^\Delta_\bullet(X;A):=C^\Delta_\bullet(X)\otimes_\mathbb{Z}A,\qquad C_\bullet(X;A):=C_\bullet(X)\otimes_\mathbb{Z}A\]we would have obtained exactly this kind of homology theory.
Most properties of homology follow from the axioms in Definition 6. For example, the generalization of equation (1), which is the goal of this post, can be derived from them. Let a topological space \(X\) be the union of two open sets \(X=U\cup V\). Applying homology to the inclusions
we obtain, by exactness, a morphism between long exact sequences; here the inclusion
\[(V,U\cap V)\rightarrow (X,U)\]induces an isomorphism in homology by the excision axiom, so all corresponding components in the above morphism are isomorphisms. That is, we obtain the following morphism of long exact sequences
Here \(i,j,k\) are the maps induced by the respective inclusions, \(\partial\) are connecting maps, and \(p\) are cokernel morphisms; indices are omitted for convenience. Denote this morphism of long exact sequences by \(\alpha\), and let \(\Cone(\alpha)\) be its mapping-cone exact sequence
\[\begin{aligned}\cdots &\overset{\overline{\partial}}{\longrightarrow} H_{n+1}(X)\oplus H_{n+1}(V, U\cap V)\overset{\overline{\Phi}}{\longrightarrow} H_{n+1}(X,U)\oplus H_n(U\cap V)\overset{\overline{\Psi}}{\longrightarrow} H_n(U)\oplus H_n(V)\\ \phantom{\cdots}&\overset{\overline{\partial}}{\longrightarrow} H_n(X)\oplus H_n(V, U\cap V)\rightarrow \cdots\end{aligned}\]By the excision result above, the long exact sequence
\[\cdots \rightarrow 0 \rightarrow H_{n+1}(V, U\cap V)\rightarrow H_{n+1}(X,U)\rightarrow 0 \rightarrow \cdots\tag{4}\]exists, and using this we can express \(\Cone(\alpha)\) as the direct sum of this trivial long exact sequence and the long exact sequence
\[\cdots \rightarrow H_{n+1}(U)\oplus H_{n+1}(V)\rightarrow H_{n+1}(X)\rightarrow H_n(U\cap V)\rightarrow H_n(U)\oplus H_n(V)\rightarrow\cdots\tag{5}\]Since both \(\Cone(\alpha)\) and (4) are exact, (5) is exact as well; the differential maps of (5) are obtained from the mapping-cone exact sequence by the change of basis determined by the isomorphisms \(i_V\). Carrying out this computation explicitly yields the following.
Proposition 7 (Mayer–Vietoris sequence) Let a topological space \(X\) be the union of two open sets \(X=U\cup V\), and let \(H\) be a homology theory on \(X\). Then there is a long exact sequence
\[\cdots \rightarrow H_{n+1}(U)\oplus H_{n+1}(V)\overset{\Psi}{\longrightarrow} H_{n+1}(X)\overset{\partial}{\longrightarrow} H_n(U\cap V)\overset{\Phi}{\longrightarrow} H_n(U)\oplus H_n(V)\rightarrow\cdots\]where the maps \(\Psi, \Phi\) are given by
\[\Psi(u,v)=u+v,\qquad \Phi(x)=(x,-x)\]In particular, for \(n=1\) we recover equation (1), which we obtained at the outset by abelianizing the Seifert–van Kampen theorem; in this sense the Mayer–Vietoris sequence may be regarded as the homology analogue of the Seifert–van Kampen theorem.
References
[Hat] A. Hatcher, Algebraic Topology. Cambridge University Press, 2022.
[May] J. P. May, A concise course in algebraic topology.
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