Definition 7.2.1.
Let \(X\) be a topological space. A singular \(n\)-simplex of \(X\) is a continuous function \(\sigma :\delta n \to X\text{.}\)
Section 7.2 Singular Homology
“...Commutative algebra is a lot like topology, only backwards.”―John Baez
Subsection 7.2.1 Definitions and Induced Homomorphisms
Definition 7.2.2.
Let \(X\) be a topological space and let \(n \geq 0\text{.}\) Let \(Sn\) be the set of singular \(n\)-simplices of \(X\text{.}\) The group of singular \(n\)-chains of \(X\) is \(Cnsing(X) = Cn(X) := \oplus singular n-simplices \sigma \Z = \oplus singular n-simplices \sigma \Z \sigma\text{.}\) An element of \(Cnsing(X)\) is called a (singular) \(n\)-chain, and written \(\sum singular n-simplices \sigma m\sigma \sigma\text{,}\) where each \(m\sigma \in \Z\) and all but finitely many \(m\sigma\)'s are \(0\text{.}\)
Definition 7.2.3.
Let \(X\) be a topological space and let \(n \geq 1\text{.}\) The \(n\)-th singular boundary map is the group homomorphism \(∂nsing = ∂nX = ∂n: Cnsing(X) \to Cn-1sing(X)\) defined by \(∂n(\sum singular n-simplices \sigma m\sigma \sigma ) := \sum singular n-simplices \sigma m\sigma ∂n(1\sigma ) and ∂n(1\sigma ) := \sum i=0n (-1)i \sigma \circ Fi(n)\text{.}\) The \(0\)-th singular boundary map is the group homomorphism \(∂0sing = ∂0: C0sing(X) \to 0\text{.}\)
Lemma 7.2.4.
If \(X\) is a topological space and \(n \geq 1\text{,}\) then \(Im ∂n+1sing \sse Ker ∂nsing\text{.}\)
Definition 7.2.5.
Let \(X\) be a topological space and let \(n \geq 0\text{.}\) The \(n\)-th singular homology group of \(X\) is the group \(Hnsing(X) = H_n(X) := Ker ∂nsing / Im ∂n+1sing\text{.}\)
Example 7.2.6.
Examples
Proposition 7.2.7.
If \(X\) is a space with exactly one point, then \(H_0(X) = \Z\) and \(Hn = 0\) for all \(n \geq 1\text{.}\)
Theorem 7.2.8.
If a space \(X\) has path components \(X_\alpha\text{,}\) then \(Hnsing(X) = \oplus \alpha Hnsing(X_\alpha )\) for all \(n \geq 0\text{.}\)
Theorem 7.2.9.
If \(X\) is a path-connected space, then \(H0sing(X) \cong \Z\text{.}\)
Excursion into homological algebra.
Definition 7.2.10.
A chain complex, denoted \(C•\text{,}\) is a sequence \(\cdots \to Cn+1 \to Cn \to Cn-1 \to \cdots C0 \to 0\) with group homomorphisms \(∂nC = ∂n: Cn \to Cn-1\) and \(∂0C = ∂0: C0 \to 0\) satisfying \(Im ∂n+1 \sse Ker ∂n\) for all \(n\text{.}\) An element of \(Ker ∂n\) is called an \(n\)-cycle and an element of \(Im ∂n+1\) is an \(n\)-boundary. For \(n \geq 0\text{,}\) the \(n\)-th homology group of \(C• is H_n(C•) := Ker ∂nC / Im ∂n+1C\text{.}\)
Definition 7.2.11.
A chain map from a chain complex \(C•\) to a chain complex \(D•\text{,}\) written \(f♯:C• \to D•\text{,}\) is a collection of group homomorphisms \(fn♯: Cn \to Dn\) for all \(n \geq 0\) satisfying \(∂nD \circ fn♯ = fn-1♯ \circ ∂nC\) for all \(n\text{.}\)
Definition 7.2.12.
Let \(f♯:C• \to D•\) be a chain map. The \(n\)-th homology homomorphism induced by \(f♯\) is the function \(fn∗: H_n(C•) \to H_n(D•)\) defined by \(fn∗(γ + Im ∂n+1C) := fn♯(γ) + Im ∂n+1D\text{.}\)
Lemma 7.2.13.
If \(f♯:C• \to D•\) is a chain map, then the \(n\)-th homology homomorphism \(fn∗\) induced by \(f♯\) is a well-defined group homomorphism.
Definition 7.2.14.
Two chain maps \(f♯,g♯: C• \to D•\) are chain homotopic, written \(f♯ \simeq g♯\text{,}\) if there is a collection of group homomorphisms \(hn: Cn \to Dn+1 \)satisfying \(gn - fn = ∂n+1D \circ hn + hn-1 \circ ∂nC\) for all \(n\text{.}\) The collection of functions \(\{hn\}\) is called a chain homotopy from \(f♯\) to \(g♯\text{.}\)
Proposition 7.2.15.
If \(f♯,g♯: C• \to D•\) are chain homotopic chain maps, then \(fn∗ = gn∗\) for all \(n\text{.}\)
Definition 7.2.16.
A chain map \(f♯: C• \to D•\) is a chain homotopy equivalence if there is a chain map \(g♯: D• \to C•\) such that \(f♯ \circ g♯ \simeq 1D•\) and \(g♯ \circ f♯ \simeq 1C•\text{.}\)
Proposition 7.2.17.
If \(f♯: C• \to D•\) is a chain homotopy equivalence, then \(fn∗: H_n(C•) \to H_n(D•)\) is an isomorphism for all \(n\text{.}\)
Theorem 7.2.18.
If \(X\) is a \(\delta\) -complex, then \(Hn\delta (X) \cong Hnsing(X)\) for all \(n \geq 0\text{.}\)
Corollary 7.2.19.
If \(X\) is a topological space with two \(\delta\) -complex structures \(X'\) and \(X''\text{,}\) then \(Hn\delta (X') \cong Hn\delta (X'')\) for all \(n \geq 0\text{.}\)
Definition 7.2.20.
Let \(X\) and \(Y\) be topological spaces and let \(f:X \to Y\) be a continuous function, and let \(n \geq 0\text{.}\) The \(n\)-th homology homomorphism induced by \(f\) is the homology homomorphism \(fn∗: H_n(X) \to H_n(Y)\) induced by the (induced) chain map \(f♯: C•sing(X) \to C•sing(Y)\) defined by \(fn♯(\sum singular n-simplices \sigma of X m\sigma \sigma ) := \sum singular n-simplices \sigma\) of \(X m\sigma f \circ \sigma\text{.}\)
Lemma 7.2.21.
If \(X\) and \(Y\) are topological spaces and \(f:X \to Y\) is a continuous function, then the induced chain map \(f♯\) is a well-defined chain map and hence the induced homology homomorphisms are well-defined homomorphisms.
Proposition 7.2.22.
- If \(f: X \to Y\) and \(g: Y \to Z\) are continuous then \((g \circ f)n∗ = gn∗ \circ fn∗\) for all \(n\text{.}\)
- \((1X)n∗ = 1H_n(X)\) for all \(n\text{.}\)
- If \(f\) and \(g\) are homotopic maps, then \(fn∗ = gn∗\) for all \(n\text{.}\)
Theorem 7.2.23.
If \(X\) and \(Y\) are homotopy equivalent spaces (that is, \(X \simeq Y\)), then \(H_n(X) \cong H_n(Y)\) for all \(n\text{.}\)
Example 7.2.24.
Examples
Corollary 7.2.25.
If \(X\) is a contractible space, then \(H_0(X) \cong \Z\) and \(H_n(X) = 0\) for all \(n \geq 1\text{.}\)
Subsection 7.2.2 Decomposing Spaces and Homology Groups
Motivation: Want an SVK theorem for homology
Another excursion into homological algebra:.
Definition 7.2.26.
A sequence of \(φ:A \to B\text{,}\) \(\theta :B \to C\) of groups and homomorphisms is exact at \(B\) if \(Im φ = Ker \theta \text{.}\) A sequence \(\cdots Dn+1 \to Dn \to \cdots \) is exact if it is exact at every group.
Lemma 7.2.27.
- A sequence \(φ:A \to B\text{,}\) \(\theta :B \to 0\) is exact if and only if \(φ\) is onto.
- A sequence \(φ:0 \to B, \theta :B \to C\) is exact if and only if \(\theta \) is one-to-one.
- If \(A \to B \to C \to D \to E\) is an exact sequence with homomorphisms \(φ:A \to B\) and \(ψ:D \to E\) satisfying \(Im φ = B\) and \(Ker ψ = 0\text{,}\) then \(C = 0\text{.}\)
Theorem 7.2.28. MV = Mayer-Vietoris Theorem.
Suppose that \(X\) is a topological space with subspaces \(A,B\) such that \(X = Int(A) \cup Int(B)\text{.}\) Then there is an exact sequence \(\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to Hn-1(A \cap B) \to \cdots \to H_0(X) \to 0\text{,}\) such that each homorphism \(φn: H_n(A \cap B) \to H_n(A) \oplus H_n(B)\) is given by \(φn := (iAB n∗,-iBA n∗)\) and each homomorphism \(ψn: H_n(A) \oplus H_n(B) \to H_n(X)\) is given by \(ψn := jA n∗ + jB n∗\text{,}\) where \(iAB: A \cap B \to A\text{,}\) \(iBA: A \cap B \to B\text{,}\) \(jA: A \to X\text{,}\) and \(jB: B \to X\) are inclusion maps.
Example 7.2.29.
Examples
Proposition 7.2.30.
\(H_i(Sn) \cong \Z\) for \(i = 0,n\text{,}\) and \(H_i(Sn) = 0\) for all \(i \ne 0,n\text{.}\)
Yet another excursion into homological algebra.
Definition 7.2.31.
A short exact sequence of chain complexes is a sequence \(0• \to A• \to B• \to C• \to 0•\) of chain complexes and chain maps such that for all \(n \geq 0\) the sequence \(0 \to An \to Bn \to Cn \to 0\) of groups and homomorphisms is exact.
Theorem 7.2.32. Zig-zag Lemma.
If \(0_• \to A_•, i♯:A_• \to B_•, j♯:B_• \to C_•, C_• \to 0\) is a short exact sequence of chain complexes, then there is a (long) exact sequence \(\cdots H_n(A) \to H_n(B) \to H_n(C) \to H_{n-1}(A) \to \cdots ,\) with homomorphisms \(in∗:H_n(A) \to H_n(B)\) and \(jn∗:H_n(B) \to H_n(C)\text{.}\)
Proof of the MV Theorem.
Definition 7.2.33.
Let \(X\) be a topological space with subspaces \(A,B\) such that \(X = Int(A) \cup Int(B)\text{.}\) The chain complex \(C_•(A+B)\) is defined by: \(C_n(A+B) := {\sum m\sigma \sigma + \sum kττ | \text{ each }\sigma ,τ \text{ is a singular} n-\text{ simplex of }X, \text{ each }\sigma (\delta n) \sse A \text{ and }τ(\delta n) \sse B, \text{ and each }m\sigma , kτ \in \Z }\) and \(∂nA+B := ∂nX|Cn(A+B)Cn-1(A+B)\text{.}\)
Proposition 7.2.34.
Let \(X\) be a topological space with subspaces \(A,B\) such that \(X = Int(A) \cup Int(B)\text{.}\) The sequence \(0_• \to C_•(A \cap B) \to C_•(A) \oplus C_•(B) \to C_•(A+B) \to 0•\text{,}\) with homomorphisms \(φ̃_n: C_n(A \cap B) \to C_n(A) \oplus C_n(B)\) given by \(φ̃_n := (i_{AB} n♯,-i_{BA} n♯) and ψ̃_n: C_n(A) \oplus C_n(B) \to C_n(A+B)\) given by \(ψ̃_n := jA n♯ + jB n♯\text{,}\) is exact.
Theorem 7.2.35. Small Chains Theorem.
Let \(X\) be a topological space with subspaces \(A,B\) such that \(X = Int(A) \cup Int(B)\text{.}\) Then \(H_n(X) \cong H_n(A+B)\) for all \(n\text{.}\)
