REMARKS IN HOMOLOGY

DANIEL VICTOR TAUSK

Date: December 1999.

1. The Cellular Chain Complex of a CW-complex

We compute the cellular chain complex of a CW-complex, without using simplicial homology. This is an example, illustrating a general technique. Our basic reference for notation and terminology is [1].

Let D denote the unit closed disc in IR^{2} ≅ ℂ and let S^{1} = Bd D denote the boundary of D. Let n ≥ 2 be a fixed integer and denote by r: D → D the map r(z) = z · e^{\frac{2πi}{n}} given by counterclockwise rotation of \frac{2π}{n}. We define an equivalence relation on D by identifying each z ∈ S^{1} with the points r^{(i)}(z), i = 1, 2, …, n − 1, where r^{(i)} denotes the i − th iteration of r. We denote the quotient space by X and call it the n-fold dunce cap. Let q: D → X denote the quotient map.

If A ⊂ D is a closed set then the saturation q^{−1}(q(A)) of A is given by

q^{−1}(q(A)) = A ∪ r(A ∩ S^{1}) ∪ … ∪ r^{(n−1)}(A ∩ S^{1}),

and therefore is also closed. It follows that the quotient map q is a closed map and since D is a T 4-space, the n-fold dunce cap X is also a T 4-space (see [1, Theorem 37.1]).

Let C ⊂ S^{1} denote the arc

C = \{e^{it} : t ∈ [0, \frac{2π}{n} ]\} .

Denote by Bd C = {1, r(1)} the boundary of C and by int C = C \Bd C the interior of C. Denote also by int D = D \ S^{1} the interior of D.

The restriction of q to C is also a quotient map onto q(C), since C is compact and q(C) is Hausdorff. Since int D and int C are saturated open sets in D and in C respectively, it follows that the restrictions of q to int D and int C are again quotient maps and, being one-to-one, are in fact homeomorphisms.

It follows that X can be made into a finite CW -complex with open cells e_{0} = {q(1)}, e_{1} = q(int C) and e_{2} = q(int D), of dimensions 0, 1 and 2 respectively. The characteristic maps^{1} for this cells are all given by appropriate restrictions of q. The skeletons of X are X^{0} = e_{0}, X^{1} = q(C) = q(S^{1}) and X^{2} = X.

We now compute the boundary operators on the cellular chain complex of X. We make first some remarks on the computation of such boundary operator for a general CW -complex X. Recall that the cellular chain complex D(X) is defined by D_{p}(X) = H_{p}(X^{p}, X^{p−1}), and the boundary operator ∂: D_{p}(X) → D_{p−1}(X) is defined as the boundary homomorphism ∂_{∗} in the long exact sequence of the triple (X^{p}, X^{p−1}, X^{p−2}). Considering the inclusion (X^{p}, X^{p−1}, ∅) → (X^{p}, X^{p−1}, X^{p−2}) and using the naturality of the long exact sequence of a triple, it follows that the boundary operator of D(X) coincides with the composite map

(1.1)  H_{p}(X^{p}, X^{p−1}) ⟶ H_{p−1}(X^{p−1}) ⟶ H_{p−1}(X^{p−1}, X^{p−2})

where the first arrow above is the boundary homomorphism of the long exact sequence of the pair (X^{p}, X^{p−1}) and the second is induced by inclusion. Since the long exact sequence of a pair in reduced homology is simply a restriction of the usual long exact sequence of the pair, it follows that, for p ≥ 1, in (1.1) we can replace H_{p−1}(X^{p−1}) by the reduced homology group H̃_{p−1}(X^{p−1}).

1A characteristic map for an open p-cell e in a CW-complex X is a continuos map f: B → X mapping the interior of B homeomorphically onto e. Here B is any closed p-cell, i.e., any topological space homeomorphic to a closed p-ball.

If e ⊂ X is an open p-cell, and f: B → X is a characteristic map for e then f_{∗}: H_{p}(B, Bd B) → H_{p}(ē, ė) is an isomorphism, where ė denotes ē\e (see [1, Lemma 39.1]). The group H_{p}(ē, ė) is therefore free abelian of rank one. Denote by z_{e} the image by inclusion H_{p}(ē, ė) → H_{p}(X^{p}, X^{p−1}) of any generator of H_{p}(ē, ė). The group H_{p}(X^{p}, X^{p−1}) is free abelian and the elements z_{e} form a basis of such group, when e runs through the p-cells of X (see [1, Theorem 39.3]).

The problem of computing the boundary homomorphism in the complex D(X) reduces to the problem of computing the boundary of each z_{e}. Naturality of the long exact sequence of a pair gives us the following commutative diagram:

(1.2)  \begin{matrix}H_{p}(X^{p}, X^{p−1}) & ⟶^{∂_{∗}} & H̃_{p−1}(X^{p−1}) \\ ^{f_{∗}}↑ & & ↑^{(f|_{Bd B})_{∗}} \\ H_{p}(B, Bd B) & ⟶_{∂_{∗}} & H̃_{p−1}(Bd B) \end{matrix}

Since B is contractible, the bottom arrow in (1.2) is an isomorphism. It follows that the boundary of z_{e} in D(X) is computed (up to sign) by taking the image under f_{∗}: H̃_{p−1}(Bd B) → H_{p−1}(X^{p−1}, X^{p−2}) of any generator of H̃_{p−1}(Bd B). In each individual problem we have to manage to express such image in terms of the basis of D_{p−1}(X).

We go back to the case where X is the n-fold dunce cap. We denote by z_{e_{i}} = z_{i} the generator of D_{i}(X) ≅ ℤ for i = 0, 1, 2. We compute the boundary operator going from lower dimension to higher dimension, and the choice of sign of the generators z_{i} is done at the appropriate moment. We can choose, for instance, z_{0} to be the homology class of the point q(1) (considered as a 0-singular simplex) in H_{0}(X^{0}). Obviously the boundary of z_{0} is zero.

Following our general program, in order to compute the boundary of z_{1} we must now choose a generator of H̃_{0}(Bd C). Since Bd C = {1, r(1)}, we can pick for instance the homology class α_{0} of the 0-singular chain r(1) − 1 (identifying, as usual, points with 0-singular simplexes). Denoting by β_{0} ∈ H_{1}(C, Bd C) the generator such that ∂_{∗}(β_{0}) = α_{0} and choosing now z_{1} as image under q_{∗} of β_{0}, we get that the boundary of z_{1} is q_{∗}(α_{0}) = 0, since q(r(1)) = q(1).

Finally, we compute the boundary of z_{2}. We have to give an appropriate description of a generator α_{1} of H_{1}(S^{1}). Set

V = {1, r(1), …, r^{(n−1)}(1)} = q^{−1}(q(1)).

Consider r as a map r: (S^{1}, V) → (S^{1}, V) and define

(1.3)  γ_{1} = ∑_{j=0}^{n−1} r_{∗}^{(j)}(i_{∗}(β_{0})) ∈ H_{1}(S^{1}, V),

where i: (C, Bd C) → (S^{1}, V) is inclusion. Considering the long exact sequence of the pair (S^{1}, V), if we prove that γ_{1} is annihilated by the boundary homomorphism ∂_{∗}: H_{1}(S^{1}, V) → H_{0}(V), it will follow that we can find α_{1} ∈ H_{1}(S^{1}) which is mapped by inclusion to γ_{1}.

Naturality of the long exact sequence of a pair together with ∂_{∗}(β_{0}) = α_{0} implies:

∂_{∗}(γ_{1}) = ∑_{j=0}^{n−1} r_{∗}^{(j)}(i_{∗}(α_{0})) = ∑_{j=0}^{n−1} [r^{(j+1)}(1) − r^{(j)}(1)] = 0,

where now the maps i, r are considered as i: Bd C → V and r: V → V.

We have thus defined an element α_{1} ∈ H_{1}(S^{1}). We now prove it is a generator. Since S^{1} \ int C is contractible, inclusion induces an isomorphism H_{1}(S^{1}) → H_{1}(S^{1}, S^{1} \ int C). The image of α_{1} under this isomorphism is equal to the image of β_{0} under inclusion (C, Bd C) → (S^{1}, S^{1} \ int C). Indeed, observe that all but the term corresponding to j = 0 in the right side of (1.3) become zero in H_{1}(S^{1}, S^{1} \ int C). Now, inclusion (C, Bd C) → (S^{1}, S^{1} \ int C) induces an homology isomorphism, as one sees by making the excision of a point in S^{1} \ int C and then using a deformation retract. Since β_{0} is a generator of H_{1}(C, Bd C), we conclude that α_{1} is a generator of H_{1}(S^{1}).

We now chose the generator β_{1} ∈ H_{2}(D, S^{1}) which maps onto α_{1} under the boundary homomorphism ∂_{∗} of the long exact sequence of (D, S^{1}). Using again the general principle of diagram (1.2), we set z_{2} = q_{∗}(β_{1}) and therefore the boundary of z_{2} is q_{∗}(α_{1}) ∈ D_{1}(X). Since q ∘ r|_{S^{1}} = q|_{S^{1}} and q_{∗}(β_{0}) = z_{1}, it follows from (1.3) that q_{∗}(α_{1}) = n · z_{1}. The cellular chain complex D(X) is therefore isomorphic to:

(1.4)  ⋯ ⟶ 0 ⟶ ℤ ⟶^{n·} ℤ ⟶^{0} ℤ ⟶ 0 ⟶ ⋯

and therefore H_{2}(X) = 0, H_{1}(X) ≅ ℤ/nℤ and H_{0}(X) ≅ ℤ.

2. Homology Classes Determined by Curves: The Hurewicz Homomorphism

We associate to each curve in a topological space a singular 1-simplex, which in appropriate situations determine a homology class. We study the relation between homotopy of curves and homology.

For p ≥ 0, we denote by Δ_{p} the standard p-simplex and by e_{0}, …, e_{p} its vertices. If v_{0}, …, v_{p} are points in an affine space V then ℓ(v_{0}, …, v_{p}) denotes the unique affine map from Δ_{p} to V which maps e_{i} to v_{i}, i = 0, 1, …, p. If X is a topological space, a singular p-simplex in X is a continuous map T: Δ_{p} → X. We identify singular 0-simplexes in X with points in X. We denote by S_{p}(X) the free abelian group generated by the singular p-simplexes in X. The elements of S_{p}(X) are called singular p-chains in X. The boundary homomorphism ∂_{p}: S_{p}(X) → S_{p−1}(X) is defined by:

∂_{p}(T) = ∑_{i=0}^{p}(−1)^{i} T ∘ ℓ(e_{0}, …, ê_{i}, …, e_{p}),

for any singular p-simplex T, where the hat marks the deleted element in the sequence. We have ∂_{p−1} ∘ ∂_{p} = 0, for every p.

The chain complex (S(X), ∂) is called the singular chain complex associated to X. We denote by Z_{p}(X) = Ker ∂_{p} the group of singular p-cycles and by B_{p}(X) = Im ∂_{p+1} the group of singular p-boundaries. As usual, H_{p}(X) = Z_{p}(X)/B_{p}(X) is the p-th singular homology group of X.

If f: X → Y is a continuous map we denote by f_{#} the homomorphism from S_{p}(X) to S_{p}(Y) which maps each singular p-simplex T in X to f ∘ T. We have that f_{#} is a chain map, i.e., commutes with the boundary homomorphism. It thus takes cycles of X to cycles of Y and boundaries of X to boundaries of Y, and defines a homomorphism f_{∗}: H_{p}(X) → H_{p}(Y) in the homology groups.

Given singular p-chains c_{1}, c_{2} ∈ S_{p}(X), we say that they are homologous if c_{1} − c_{2} ∈ B_{p}(X) (we do not require c_{1} and c_{2} to be cycles when this terminology is used). Obviously in this case c_{1} is a cycle if and only if c_{2} is a cycle. Given a singular p-chain c, we denote by c + B_{p}(X) its equivalence class in S_{p}(X)/B_{p}(X). If c is a singular p-cycle then c + B_{p}(X) ∈ H_{p}(X) is called the homology class determined by c.

Let γ: [a, b] → X be a (continuous) curve in a topological space X. We set:

𝔥(γ) = γ_{#}(ℓ(a, b)) = γ ∘ ℓ(a, b).

Then 𝔥(γ) is a singular 1-simplex in X whose boundary is γ(b) − γ(a). If γ is a loop, i.e., if γ(a) = γ(b) then 𝔥(γ) is a cycle and defines a homology class in H_{1}(X). Remark 2.1. If γ has endpoints in a subspace A ⊂ X then 𝔥(γ) is a relative cycle in (X, A), i.e., ∂(𝔥(γ)) ∈ S_{1}(A). It defines therefore a relative homology class in (X, A). More in general, if B ⊂ A is such that inclusion i: (X, B) → (X, A) induces an isomorphism on the first homology group then we can apply (i_{∗})^{−1} to the homology class of 𝔥(γ) in (X, A) to associate to γ an homology class in (X, B).

We prove some basic properties of singular 1-chains determined by curves. Proposition 2.2. Let γ: [a, b] → X be a curve and let σ: [c, d] → [a, b] be a continuous map. If σ(c) = a and σ(d) = b then 𝔥(γ ∘ σ) is homologous to 𝔥(γ). If σ(c) = b and σ(d) = a then 𝔥(γ ∘ σ) is homologous to −𝔥(γ). If r ∈ (a, b) then 𝔥(γ) is homologous to 𝔥(γ|_{[a,r]}) + 𝔥(γ|_{[r,b]}). Finally, if γ is a constant map then 𝔥(γ) is homologous to zero.

Proof. Since [a, b] is acyclic, every p-cycle with p ≥ 1 is a p-boundary. It follows that two p-chains in [a, b] are homologous if and only if they have the same boundary (p ≥ 1). It now follows easily that σ ∘ ℓ(c, d) is homologous to ℓ(a, b) if σ(c) = a and σ(d) = b. If σ(c) = b and σ(d) = a then σ ∘ ℓ(c, d) is homologous to −ℓ(a, b). Moreover, ℓ(a, r) + ℓ(r, b) is homologous to ℓ(a, b) and any constant singular 1simplex in [a, b] is homologous to zero. The proposition follows by observing that, since γ_{#} is a chain map, if c_{1} and c_{2} are homologous in [a, b] then γ_{#}(c_{1}) and γ_{#}(c_{2}) are homologous in X. □

We’ll say that two curves γ, μ: [a, b] → X are homotopic if there they are homotopic with fixed endpoints, i.e., if there exists a continuous map H: [a, b] × [0, 1] → X with H(s, 0) = γ(s), H(s, 1) = μ(s), H(a, t) = γ(a) = μ(a) and H(b, t) = γ(b) = μ(b) for every t ∈ [0, 1] and s ∈ [a, b]. The map H is called a homotopy from γ to μ. Homotopic curves define homologous chains: Proposition 2.3. If γ, μ: [a, b] → X are homotopic then 𝔥(γ) and 𝔥(μ) are homologous.

Proof. Let H: [a, b] × [0, 1] → X be a homotopy from γ to μ. Consider the singular 1-chain

(2.1)  c = ℓ((a, 0), (b, 0)) − ℓ((a, 1), (b, 1)) − ℓ((a, 0), (a, 1)) + ℓ((b, 0), (b, 1))

in [a, b]×[0, 1]. Since ∂c = 0 and [a, b]×[0, 1] is acyclic, we see that c is a 1-boundary. Since H_{#} is a chain map, H_{#}(c) is a 1-boundary as well.

Applying H_{#} to both sides of (2.1), the first two terms on the righthand side give γ_{#} −μ_{#} while the last two give singular chains determined by constant curves. By Proposition 2.2 those are boundaries and the proof is complete. □

Let Ω(X) denote the set of all curves γ: [0, 1] → X. We denote by [γ] the equivalence class of curves homotopic to γ and by \overline{Ω}(X) the quotient of Ω(X) consisting of the homotopy classes [γ], γ ∈ Ω(X). If γ, μ ∈ Ω(X) are such that γ(1) = μ(0), we define the concatenation of γ and μ by:

\begin{cases}(γ · μ)(s) = γ(2s), & s ∈ [0, \frac{1}{2}] , \\ (γ · μ)(s) = μ(2s − 1), & s ∈ [\frac{1}{2}, 1] . \end{cases}

Concatenation is a (partially defined) binary operation in Ω(X). It’s not hard to see that it induces a (partially defined) binary operation in \overline{Ω}(X), i.e., if [γ] = [γ′] and [μ] = [μ′] then [γ · μ] = [γ′ · μ′]. It’s also not hard to see that concatenation is associative in \overline{Ω}(X) (whenever all products involved are defined).

The homotopy classes of constant curves work as neutral elements for concatenation. More precisely if, for x_{0} ∈ X, 𝔠_{x_{0}} : [0, 1] → X denote the constant map 𝔠_{x_{0}}(s) = x_{0}, s ∈ [0, 1] then:

(2.2)  [𝔠_{γ(0)}] · [γ] = [γ] and [γ] · [𝔠_{γ(1)}] = [γ],

for every γ ∈ Ω(X). We also set:

γ^{−1}(s) = γ(1 − s), s ∈ [0, 1],

for every γ ∈ Ω(X). Then, it’s not hard to show that:

(2.3)  [γ] · [γ^{−1}] = [𝔠_{γ(0)}] and [γ^{−1}] · [γ] = [𝔠_{γ(1)}].

Moreover, the transformation γ ↦ γ^{−1} preserves homotopy classes, so that we can define a map

\overline{Ω}(X) ∋ [γ] ↦ [γ]^{−1} = [γ^{−1}] ∈ \overline{Ω}(X).

It’s easily seen that:

(2.4)  ([γ] · [μ])^{−1} = [μ]^{−1} · [γ]^{−1}.

If f: X → Y is continuous then we associate to it a map f_{#} : \overline{Ω}(X) → \overline{Ω}(Y) such that

f_{#}([γ]) = [f ∘ γ],

for every γ ∈ Ω(X). It’s easily seen that f_{#} is well-defined and that:

\begin{matrix}f_{#}([γ] · [μ]) = & f_{#}([γ]) · f_{#}([μ]), f_{#}([γ]^{−1}) = f_{#}([γ])^{−1} and \\ f_{#}([𝔠_{x_{0}}]) = [𝔠_{f(x_{0})}], \end{matrix} ; (2.5)

for every γ, μ ∈ Ω(X) with γ(1) = μ(0) and every x_{0} ∈ X.

Recall that we also denote by f_{#} the chain map from S(X) to S(Y), but the context should distinguish between the two meanings of f_{#}.

Relating homotopy classes with homology classes, we define a map

Θ: \overline{Ω}(X) → S_{1}(X)/B_{1}(X)

such that for every γ ∈ Ω(X):

Θ([γ]) = 𝔥(γ) + B_{1}(X)

Observe that Θ([γ]) is in general not a homology class, unless γ is a loop. Proposition 2.3 says that Θ is well-defined and Proposition 2.2 implies that:

(2.6) Θ([γ] · [μ]) = Θ([γ]) + Θ([μ]), Θ([γ]^{−1}) = −Θ([γ]) and Θ([𝔠_{x_{0}}]) = 0,

for every γ, μ ∈ Ω(X) with γ(1) = μ(0) and every x_{0} ∈ X. Moreover, if f: X → Y is a continuous map, it’s easily seen that the following diagram commutes:

\overline{Ω}(X)  ⟶^{f_{#}}  \overline{Ω}(Y)
Θ↓  ↓^{Θ}
S_{1}(X)/B_{1}(X) ⟶_{f̄_{#}} S_{1}(Y)/B_{1}(Y)

where f̄_{#} is induced by f_{#} : S_{1}(X) → S_{1}(Y).

Let x_{0} ∈ X. We denote by π_{1}(X, x_{0}) the subset of \overline{Ω}(X) consisting of homotopy classes of curves γ ∈ Ω(X) such that γ(0) = γ(1) = x_{0}. Then concatenation restricts to a binary operation in π_{1}(X, x_{0}) which makes it a group. The neutral element is [𝔠_{x_{0}}] and the inverse of [γ] is [γ^{−1}]. We call π_{1}(X, x_{0}) the fundamental group of X with base point x_{0}. The map Θ restricts to a map from π_{1}(X, x_{0}) to H_{1}(X), which is a group homomorphism by (2.6). This restriction (also denoted by Θ) is called the Hurewicz homomorphism. The map f_{#} induced by a continuos map f: X → Y restricts to a group homomorphism from π_{1}(X, x_{0}) to π_{1}(Y, f(x_{0})), by (2.5).

We prove now the main theorem of the section. Theorem 2.4. Let X be a path connected topological space and let x_{0} ∈ X. Then the Hurewicz homomorphism Θ: π_{1}(X, x_{0}) → H_{1}(X) is surjective and its kernel is the commutator subgroup π_{1}(X, x_{0})′ of π_{1}(X, x_{0}).

Proof. Let G denote the quotient group π_{1}(X, x_{0})/π_{1}(X, x_{0})′. Since π_{1}(X, x_{0})′ is the commutator subgroup of π_{1}(X, x_{0}), G is abelian (althought we shall write its group operation multiplicatively). Since H_{1}(X) is abelian, Θ vanishes on π_{1}(X, x_{0})′ and therefore induces a homomorphism \overline{Θ}: G → H_{1}(X) such that \overline{Θ}∘q = Θ|_{π_{1}(X,x_{0})}, where q: π_{1}(X, x_{0}) → G is the quotient map. To prove the theorem, we have to show that \overline{Θ} is an isomorphism.

For each x ∈ X choose a curve η_{x}: [0, 1] → X with η_{x}(0) = x_{0} and η_{x}(1) = x. Define a homomorphism ψ: S_{1}(X) → G by:

ψ(γ) = q([η_{γ(0)}] · [γ] · [η_{γ(1)}]^{−1}),

for every γ ∈ Ω(X). Here we identify the standard 1-simplex Δ_{1} with the unit interval [0, 1] (by the affine isomorphism ℓ(0, 1)), so that a curve γ: [0, 1] → X is the same as a singular 1-simplex in X. Since the singular 1-simplexes form a basis for S_{1}(X) as a free abelian group and G is abelian, we have that ψ is well-defined.

We claim that ψ maps B_{1}(X) to the neutral element of G. We just have to show that ψ ∘ ∂ maps any singular 2-simplex T: Δ_{2} → X to the neutral element of G. We write:

(2.7)  ∂T = γ_{0} − γ_{1} + γ_{2},

where γ_{0} = T ∘ ℓ(e_{1}, e_{2}), γ_{1} = T ∘ ℓ(e_{0}, e_{2}) and γ_{2} = T ∘ ℓ(e_{0}, e_{1}).

Aplying ψ to both sides of (2.7), using that q, ψ are homomorphisms and using equations (2.2), (2.3), (2.4) and (2.5) we get:

\begin{aligned}ψ(∂T) & = q([η_{T (e_{1})}] · [γ_{0}] · [γ_{1}]^{−1} · [γ_{2}] · [η_{T (e_{1})}]^{−1}) \\ & = q([η_{T (e_{1})}] · T_{#}([ρ]) · [η_{T (e_{1})}]^{−1}), \end{aligned}

where ρ = (ℓ(e_{1}, e_{2}) · ℓ(e_{2}, e_{0})) · ℓ(e_{0}, e_{1}). Since Δ_{2} is contractible, [ρ] = [𝔠_{e_{1}}] and we get ψ(∂T) = q([𝔠_{x_{0}}]), which proves the claim.

It follows now that we can define a homomorphism

\overline{ψ}: S_{1}(X)/B_{1}(X) → G

by \overline{ψ}(c + B_{1}(X)) = ψ(c), for any c ∈ S_{1}(X).

Using (2.6), we compute \overline{Θ} ∘ ψ:

\begin{aligned}(\overline{Θ} ∘ ψ)(γ) & = Θ([η_{γ(0)}]) + Θ([γ]) − Θ([η_{γ(1)}]) \\ & = 𝔥(η_{γ(0)}) + 𝔥(γ) − 𝔥(η_{γ(1)}) + B_{1}(X), \end{aligned} ; (2.8)

for any singular 1-simplex γ: [0, 1] → X.

Define a homomorphism ϕ: S_{0}(X) → S_{1}(X) by ϕ(x) = 𝔥(η_{x}) for every point (singular 0-simplex) x ∈ X. By our identification of Δ_{1} with [0, 1] we have 𝔥(γ) = γ; then (2.8) implies:

(2.9)  \overline{Θ} ∘ ψ = p ∘ (Id − ϕ ∘ ∂),

where p: S_{1}(X) → S_{1}(X)/B_{1}(X) is the quotient map and Id is the identity map. Restricting (2.9) to Z_{1}(X) and passing to homology classes we get:

\overline{Θ} ∘ \overline{ψ}|_{H_{1}(X)} = Id.

We compute now \overline{ψ} ∘ \overline{Θ}:

(\overline{ψ} ∘ \overline{Θ})(q([γ])) = ψ(γ) = q([η_{x_{0}}] · [γ] · [η_{x_{0}}]^{−1}),

for any γ: [0, 1] → X with γ(0) = γ(1) = x_{0}. Since q is a homomorphism and G is abelian, we get:

\overline{ψ} ∘ \overline{Θ} = Id,

which proves that \overline{ψ}|_{H_{1}(X)} and \overline{Θ} are mutually inverse isomorphisms. □

3. Homology of Filtered Spaces

We associate a chain complex to a filtered space. Under suitable hypothesis, we show that the homology of this complex is naturally isomorphic to the singular homology of the space. The hypotheses of the theorems in this section are satisfied when one considers the case of filtrations consisting of CW-complexes and its skeletons (see the remarks at the end of the section).

A filtered space is a topological space X endowed with a sequence (X^{p})_{p∈IN} of subspaces such that for any p = 0, 1, … we have X^{p} ⊂ X^{p+1}. For simplicity we set X^{p} = ∅ for p < 0. The sequence (X^{p})_{p∈IN} is called a filtration for X.

Let X be a filtered space and let (S(X), ∂) denote its singular chain complex (see Section 2 for definitions). For p ∈ ℤ, we set:

\begin{matrix}Z_{p}(X) = Z_{p}(X^{p}, X^{p−1}) = ∂_{p}^{−1}(S_{p−1}(X^{p−1})) and \\ B_{p}(X) = B_{p}(X^{p}, X^{p−1}) = B_{p}(X^{p}) + S_{p}(X^{p−1}). \end{matrix}

As usual, H_{p}(X^{p}, X^{p−1}) = Z_{p}(X)/B_{p}(X) is the p-th relative homology group of (X^{p}, X^{p−1}). It is easily seen that Z(X) and B(X) are subcomplexes of S(X). We also set:

D(X) = Z(X)/B(X).

Then D(X) is a chain complex with the boundary operator induced by Z(X). It’s not hard to see that it coincides with the boundary homomorphism of the long exact sequence of the triple (X^{p}, X^{p−1}, X^{p−2}):

D_{p}(X) = H_{p}(X^{p}, X^{p−1}) ⟶^{∂_{∗}} H_{p−1}(X^{p−1}, X^{p−2}) = D_{p−1}(X)

Let ε: S_{0}(X) → ℤ denote the standard augmentation homomorphism, which maps every singular 0-simplex to 1 ∈ ℤ. If X^{0} is not empty, since Z_{0}(X) = S_{0}(X^{0}), we see that ε restricts to an augmentation for Z(X). Moreover, since ε vanishes on B_{0}(X) = B_{0}(X^{0}), we see that ε also defines an augmentation in the quotient complex D(X).

If X, Y are filtered spaces, we say that a map f: X → Y is filtration preserving if it is continuous and f(X^{p}) ⊂ Y^{p} for any p. In this case, the chain map f_{#} maps Z(X) to Z(Y) and B(X) to B(Y). It induces also a chain map in D(X). Observe that the map D_{p}(X) → D_{p}(Y) induced by f_{#} coincides with the homomorphism induced by f on relative homology H_{p}(X^{p}, X^{p−1}) → H^{p}(Y^{p}, Y^{p−1}). The chain maps induced by f_{#} in Z(X) and D(X) preserve augmentation if X^{0} is not empty.

A filtration (X^{p})_{p∈IN} for a space X is said to be a good filtration if the following conditions are satisfied:

• For any p and any i ≠ p, H_{i}(X^{p}, X^{p−1}) = 0.

• For any compact set K ⊂ X, there exists p such that K ⊂ X^{p}. A space X endowed with a good filtration is called a well-filtered space. Our

objective is to show that for well-filtered spaces, the homology of D(X) is naturally isomorphic to the singular homology of X. We start with two lemmas. Lemma 3.1. Let X be a well-filtered space. Then, for any p, inclusion induces an isomorphism H_{p}(X^{p+1}) → H_{p}(X).

Proof. The long exact sequence of the pair (X^{i+1}, X^{i}) shows that

H_{p+1}(X^{i+1}, X^{i}) ⟶^{∂_{∗}} H_{p}(X^{i}) ⟶ H_{p}(X^{i+1}) ⟶ H_{p}(X^{i+1}, X^{i})

is exact, where the unlabeled arrows are induced by inclusion. For i ≥ p + 1 we conclude that inclusion induces an isomorphism H_{p}(X^{i}) → H_{p}(X^{i+1}). Therefore inclusion also induces an isomorphism H_{p}(X^{i}) → H_{p}(X^{j}), for j ≥ i ≥ p + 1.

Now let α ∈ H_{p}(X) be a homology class. There exists a compact set K ⊂ X such that α is in the image of H_{p}(K) → H_{p}(X). We can then find j ≥ p + 1 such that K ⊂ X^{j}. Then α is also in the image of H_{p}(X^{j}) → H_{p}(X), and therefore in the image of H_{p}(X^{p+1}) → H_{p}(X). We have proved that H_{p}(X^{p+1}) → H_{p}(X) is surjective.

For injectivity, let α ∈ H_{p}(X^{p+1}) be a homology class whose image under H_{p}(X^{p+1}) → H_{p}(X) vanishes. Then we can find a compact set K ⊂ X such that α also vanishes under H_{p}(X^{p+1}) → H_{p}(X^{p+1} ∪ K). Again, there exists j ≥ p + 1 such that K ⊂ X^{j} and therefore α vanishes under H_{p}(X^{p+1}) → H_{p}(X^{j}) as well. This last map is an isomorphism and we get α = 0. □

Lemma 3.2. Let X be a well-filtered space. Then for all p, i with i < p we have H_{p}(X^{i}) = 0.

Proof. For j ≤ i, the long exact sequence of the triple (X^{i}, X^{j}, X^{j−1}) shows that

H_{p}(X^{j}, X^{j−1}) ⟶ H_{p}(X^{i}, X^{j−1}) ⟶ H_{p}(X^{i}, X^{j}) ⟶^{∂_{∗}} H_{p−1}(X^{j}, X^{j−1})

is exact, where the unlabeled arrows are induced by inclusion. For j ≤ p − 2 we conclude that inclusion induces an isomorphism H_{p}(X^{i}, X^{j−1}) → H_{p}(X^{i}, X^{j}). Therefore, since X_{k} = ∅ for k < 0, inclusion also induces an isomorphism H_{p}(X^{i}) → H_{p}(X^{i}, X^{j}), where j = min{i, p − 2}.

If i ≤ p − 2 we have j = i, so that H_{p}(X^{i}, X^{j}) = 0 and the proof is complete. Otherwise, i = p − 1 and H_{p}(X^{i}, X^{j}) = H_{p}(X^{p−1}, X^{p−2}) = 0 and the proof is complete as well. □

We can now prove our two main theorems. Theorem 3.3. Let X be a well-filtered space. Then inclusion Z(X) → S(X) induces an isomorphism in homology. If X^{0} is not empty, the same holds in reduced homology. The isomorphism is natural, i.e., if f: X → Y is filtration preserving then the diagram

(3.1)  \begin{matrix}H(X) & ⟶^{f_{∗}} & H(Y) \\ ↑ & & ↑ \\ H(Z(X)) & ⟶^{f_{∗}} & H(Z(Y)) \end{matrix}

commutes, where the vertical arrows are induced by inclusion and the bottom arrow is induced by f_{#} : Z(X) → Z(Y).

Proof. For any p, the p-th cycle group of Z(X) is Z_{p}(X^{p}) and the p-th boundary group of Z(X) is B_{p}(X^{p+1}) ∩ Z_{p}(X^{p}). We have to prove that inclusion induces an isomorphism:

\frac{Z_{p}(X^{p})}{B_{p}(X^{p+1}) ∩ Z_{p}(X^{p})} ⟶ \frac{Z_{p}(X)}{B_{p}(X)}.

By Lemma 3.1, inclusion induces an isomorphism:

(3.2)  \frac{Z_{p}(X^{p+1})}{B_{p}(X^{p+1})} ⟶ \frac{Z_{p}(X)}{B_{p}(X)}.

The long exact sequence of the pair (X^{p+1}, X^{p}) shows that

H_{p}(X^{p}) ⟶ H_{p}(X^{p+1}) ⟶ H_{p}(X^{p+1}, X^{p}) = 0

is exact, where the arrows are induced by inclusion. Therefore inclusion induces a surjective map:

\frac{Z_{p}(X^{p})}{B_{p}(X^{p})} ⟶ \frac{Z_{p}(X^{p+1})}{B_{p}(X^{p+1})},

which implies that Z_{p}(X^{p+1}) = Z_{p}(X^{p}) + B_{p}(X^{p+1}). By standard homomorphism theorems, inclusion induces an isomorphism:

(3.3)  \frac{Z_{p}(X^{p})}{B_{p}(X^{p+1}) ∩ Z_{p}(X^{p})} ⟶ \frac{Z_{p}(X^{p}) + B_{p}(X^{p+1})}{B_{p}(X^{p+1})} = \frac{Z_{p}(X^{p+1})}{B_{p}(X^{p+1})}.

Since both (3.2) and (3.3) are isomorphisms, the proof of the first part of the statement is complete. If X^{0} is not empty, inclusion Z(X) → S(X) is augmentation preserving and therefore also induces an isomorphism in reduced homology. Diagram (3.1) already commutes on the chain level. □

Theorem 3.4. Let X be a well-filtered space. Then the quotient map Z(X) → D(X) induces an isomorphism in homology. The same holds in reduced homology if X^{0} is not empty. The isomorphism is natural, i.e., if f: X → Y is filtration preserving then the diagram

(3.4)  \begin{matrix}H(Z(X)) & ⟶^{f_{∗}} & H(Z(Y)) \\ ↓ & & ↓ \\ H(D(X)) & ⟶^{f_{∗}} & H(D(Y)) \end{matrix}

commutes, where the vertical arrows are induced by the quotient map and the horizontal arrows are induced by appropriate restrictions of f_{#}.

Proof. We have a short exact sequence of chain complexes:

0 ⟶ B(X) ⟶ Z(X) ⟶ D(X) ⟶ 0.

The corresponding long exact homology sequence, shows that Z(X) → D(X) induces an homology isomorphism if and only if B(X) has zero homology.

For any p, the p-th cycle group of B(X) is B_{p}(X^{p}) + Z_{p}(X^{p−1}) and the p-th boundary group of B(X) is B_{p}(X^{p}). We have to show that Z_{p}(X^{p−1}) ⊂ B_{p}(X^{p}). By Lemma 3.2, we have H_{p}(X^{p−1}) = 0 so that:

Z_{p}(X^{p−1}) = B_{p}(X^{p−1}) ⊂ B_{p}(X^{p}).

This concludes the proof of the first part of the theorem. If X^{0} is not empty, the quotient map Z(X) → D(X) is augmentation preserving and therefore induces an isomorphism in reduced homology as well. Diagram (3.4) commutes already on the chain level. □

We wish now to generalize Theorems 3.3 and 3.4 to relative homology. We start with some definitions.

A filtered pair of spaces is a pair (X, A) of spaces such that both X and A are filtered and inclusion A → X is filtration preserving, i.e., A^{p} ⊂ X^{p} for all p. If (X, A) and (Y, B) are filtered pair of spaces, a filtration preserving map is a map f: (X, A) → (Y, B) such that f is filtration preserving both as a map X → Y and as a map A → B, i.e., if f(X^{p}) ⊂ Y^{p} and f(A^{p}) ⊂ B^{p} for all p.

If (X, A) is a filtered pair of spaces then Z(A) and B(A) are subcomplexes of Z(X) and B(X) respectively. We set:

Z(X, A) = \frac{Z(X)}{Z(A)}, B(X, A) = \frac{B(X)}{B(A)}.

As usual, we also write S(X, A) = S(X)/S(A). Recall that the homology of the chain complex S(X, A) is the relative singular homology of the pair (X, A).

Inclusion induces a (not necessarily injective) chain map D(A) → D(X). We denote its cokernel^{2} by D(X, A).

2Recall that the cokernel of a group homomorphism ϕ: G → H is the quotient H/ Im ϕ. The cokernel of a chain map ϕ is the chain complex whose p-th group is the cokernel of ϕ_{p}.

Remark 3.5. We can also define D(X, A) alternatively as the cokernel of the map B(X, A) → Z(X, A) induced by inclusion. This can be seen by applying the serpent Lemma (see [1, Exercise 2, §24]) to the diagram:

0 ⟶ B(X) ⟶ Z(X) ⟶ D(X) ⟶ 0
↑  ↑  ↑
0 ⟶ B(A) ⟶ Z(A) ⟶ D(A) ⟶ 0

where all the arrows are induced by inclusion.

Given a filtration preserving map f: (X, A) → (Y, B) then f_{#} induces chain maps Z(X, A) → Z(Y, B), B(X, A) → B(Y, B) and D(X, A) → D(Y, B). We say that (X, A) is a well-filtered pair of spaces if the following conditions hold:

• (X, A) is a filtered pair of spaces.

• Both X and A are well-filtered.

• Inclusion induces an injective chain map D(A) → D(X). We can now generalize Theorems 3.3 and 3.4 to relative homology.

Theorem 3.6. Let (X, A) be a well-filtered pair of spaces. Then the chain maps Z(X, A) → S(X, A) and Z(X, A) → D(X, A) induced by inclusion and by the quotient map respectively induce homology isomorphisms. This isomorphisms are natural, i.e., given a filtration preserving map f: (X, A) → (Y, B) then the following diagrams commute:

(3.5) \begin{matrix}H(X, A) & ⟶^{f_{∗}} & H(Y, B) \\ ↑ & & ↑ \\ H(Z(X, A)) & ⟶^{f_{∗}} & H(Z(Y, B)) \end{matrix}  \begin{matrix}H(Z(X, A)) & ⟶^{f_{∗}} & H(Z(Y, B)) \\ ↓ & & ↓ \\ H(D(X, A)) & ⟶^{f_{∗}} & H(D(Y, B)) \end{matrix}

where the horizontal arrows are induced by appropriate restrictions of f_{#} and the vertical arrows are induced by inclusions in the left diagram and by quotient maps on the right diagram.

Proof. We have a commutative diagram of chain complexes and chain maps:

(3.6)  \begin{aligned}0 & ⟶ & S(A) & ⟶ & S(X) & ⟶ & S(X, A) & ⟶ & 0 \\ & & ↑ & & ↑ & & ↑ \\ 0 & ⟶ & Z(A) & ⟶ & Z(X) & ⟶ & Z(X, A) & ⟶ & 0 \end{aligned}

where all arrows are induced by inclusions. The lines are short exact sequences; the left and middle vertical arrows induce homology isomorphisms, by Theorem 3.3. Considering the corresponding commutative diagram whose lines are long exact homology sequences and applying the five Lemma we conclude that the right vertical arrow in (3.6) induce a homology isomorphism as well.

To prove that Z(X, A) → D(X, A) induce an homology isomorphism, we proceed analogously. Simply replace S by D in diagram (3.6) and apply Theorem 3.4, together with the five Lemma in the corresponding diagram with long exact homology sequences in the lines (observe that here we use that D(A) → D(X) is injective). This completes the proof of the first part of the theorem.

Naturality follows by observing that diagrams (3.5) already commute on the chain level. □

We make some final remarks concerning cohomology. We know that if a chain map between free chain complexes induces homology isomorphisms then it also induces cohomology isomorphisms (see [1, Theorem 45.5]). The chain complexes Z(X) and S(X) are free, so that Theorem 3.3 implies that inclusion also induces a cohomology isomorphism H^{p}(X; G) → H^{p}(Z(X); G), for any p and any coefficient group G, if X is well-filtered. This isomorphism is natural, since (3.1) commutes on the chain level.

If (X, A) is a well-filtered pair of spaces, then inclusion Z(X, A) → S(X, A) also induces natural cohomology isomorphisms. Naturality follows again from the fact that the left diagram in (3.5) commutes on the chain level. To prove the isomorphism in cohomology, we just have to check that Z(X, A) and S(X, A) are free. We know that S(X, A) is free, since S(A) is spanned by a subset of the usual basis of S(X). We claim that Z(X, A) is free.

Let p ∈ ℤ be fixed. We have:

Z_{p}(A) = Z_{p}(A^{p}, A^{p−1}) ⊂ S_{p}(A^{p}) ⊂ S_{p}(X).

The map S_{p}(A^{p}) → S_{p−1}(A^{p}, A^{p−1}) given by the composition of the boundary operator and the quotient map has Z_{p}(A) as kernel and free image, which implies that Z_{p}(A) is a direct summand of S_{p}(A^{p}). Since S_{p}(A^{p}) is a direct summand of S_{p}(X), we conclude that Z_{p}(A) is a direct summand in S_{p}(X). But Z_{p}(A) ⊂ Z_{p}(X) implies that Z_{p}(A) is also a direct summand in Z_{p}(X) and therefore Z_{p}(X)/Z_{p}(A) is free. This proves the claim.

If X is well-filtered, using Theorem 3.4 and the fact that (3.4) commutes on the chain level, we get that, under the additional hypothesis that D(X) is free, the quotient map Z(X) → D(X) induces a natural cohomology isomorphism. Analogously, considering Theorem 3.6 and diagram (3.5), if (X, A) is a well-filtered pair of spaces and assuming that D(X, A) is free, the map Z(X, A) → D(X, A) induces a natural cohomology isomorphism. Unfortunately we can not prove in general that D(X) (or D(X, A)) is free.

Finally, we observe that if X is a CW-complex and if X^{p} is its p-th skeleton then X is well-filtered and D(X) is free (see [1, Theorem 39.3] and observe also that every compact subset of X is contained in a finite subcomplex of X). Moreover, if A ⊂ X is a subcomplex and A^{p} is its p-th skeleton then A^{p} = A ∩ X^{p}, (X, A) is a well-filtered pair of spaces and D(X, A) is free (this follows by considering the specific basis for D(X) described in [1, Theorem 39.3]). In this context, filtration preserving maps are also called cellular maps.

4. Associativity of Relative Cup Products

Let X be a topological space and let R be a ring. We denote as usual the singular chain complex of X by (S(X), ∂). If G is an abelian group we denote by Hom(G, R) the set of group homomorphisms ϕ: G → R, endowed with the sum given by (ϕ_{1} + ϕ_{2})(g) = ϕ_{1}(g) + ϕ_{2}(g) and left scalar multiplication by R given by (rϕ)(g) = r(ϕ(g)), for any ϕ_{1}, ϕ_{2}, ϕ ∈ Hom(G, R) and r ∈ R. Then Hom(G, R) is a left R-module. If f: G → H is a group homomorphism we denote by f^{t} the transpose of f which is the R-module homomorphism

f^{t}: Hom(H, R) ⟶ Hom(G, R)

given by f^{t}(ϕ) = ϕ∘f. Then Hom(·, R) is a contravariant functor from the category of abelian groups and group homomorphisms to the category of R-moduli and Rmodule homomorphisms. We set:

S^{p}(X; R) = Hom(S_{p}(X), R) and δ_{p} = ∂_{p+1}^{t},

so that (S·(X; R), δ) becomes a cochain complex of R-moduli, which we call the singular cochain complex of X with coefficients in R. If c^{p} ∈ S^{p}(X; R) and c_{p} ∈ S_{p}(X) we denote the value of c^{p} in c_{p} by ⟨c^{p}, c_{p}⟩ ∈ R, instead of the more usual functional notation c^{p}(c_{p}).

The groups S^{p}(X; R), Z^{p}(X; R) = Ker δ_{p} and B^{p}(X; R) = Im δ_{p−1} are called the groups of p-cochains, p-cocycles and p-coboundaries of X (with coefficients in R), respectively. The quotient H^{p}(X; R) = Z^{p}(X; R)/B^{p}(X; R) is called the p-th singular cohomology group of X with coefficients in R.

Given c^{p} ∈ S^{p}(X; R), c^{q} ∈ S^{q}(X; R) we define the cup product c^{p} ∪ c^{q} ∈ S^{p+q}(X; R) of c^{p} and c^{q} by:

(4.1)  ⟨c^{p} ∪ c^{q}, T ⟩ = ⟨c^{p}, T ∘ ℓ(e_{0}, …, e_{p})⟩ ⟨c^{q}, T ∘ ℓ(e_{p}, …, e_{p+q})⟩,

for any singular p + q-simplex T in X (the notation ℓ was introduced in Section 2). Then, for any p, q ∈ ℤ, (4.1) defines a ℤ-bilinear operator

∪: S^{p}(X; R) × S^{q}(X; R) ⟶ S^{p+q}(X; R)

which is also R-bilinear if R is commutative. If R has a unit element 1 ∈ R then the singular 0-cochain whose value is 1 on every singular 0-simplex of X is a neutral element for ∪.

The operation ∪ is associative and satisfies the following formula relatively to the coboundary operator δ (see [1, Theorem 48.1]):

(4.2)  δ(c^{p} ∪ c^{q}) = δc^{p} ∪ c^{q} + (−1)^{p}c^{p} ∪ δc^{q}.

It follows that the cup product of cocycles is a cocycle and that if both c^{p}, c^{q} are cocycles and one of them is a coboundary then c^{p} ∪ c^{q} is a coboundary. This implies that ∪ induces an operation in cohomology, which is associative, ℤ-bilinear and R-bilinear if R is commutative.

Our objective is to generalize cup products to relative cohomology and to prove its associativity, under suitable hypothesis. We introduce some notation. If A ⊂ X, we denote by S^{p}(X, A; R) the R-submodule of S^{p}(X; R) consisting of singular pcochains which vanish on S_{p}(A). There is a natural identification of S^{p}(X, A; R) with Hom(S_{p}(X, A), R), where S_{p}(X, A) = S_{p}(X)/S_{p}(A). Then S·(X, A; R) is a subcomplex of S·(X; R). More in general, given subspaces A_{1}, …, A_{n} ⊂ X, we set:

S^{p} (X, ∑_{i=1}^{n} A_{i}; R) = S^{p}(X, A_{1} + ⋯ + A_{n}; R) = ⋂_{i=1}^{n} S^{p}(X, A_{i}; R).

Then S^{p}(X, ∑_{i} A_{i}; R) is the group of p-cochains which vanish on ∑_{i=1}^{n} S_{p}(A_{i}) = S_{p}(A_{1}) + ⋯ + S_{p}(A_{n}). There is a natural identification:

S^{p} (X, ∑_{i=1}^{n} A_{i}; R) ≅ Hom (\frac{S_{p}(X)}{∑_{i=1}^{n} S_{p}(A_{i})}, R) .

We denote by Z^{p}(X, ∑_{i} A_{i}; R), B^{p}(X, ∑_{i} A_{i}; R) and H^{p}(X, ∑_{i} A_{i}; R) the p-th group of cocyles, coboundaries and cohomology of S·(X, ∑_{i} A_{i}; R) respectively.

Observe that Z^{p}(X, ∑_{i} A_{i}; R) is simply given by:

Z^{p} (X, ∑_{i=1}^{n} A_{i}; R) = Z^{p}(X; R) ∩ S^{p} (X, ∑_{i=1}^{n} A_{i}; R) ,

but the above formula doesn’t hold if we replace Z^{p} by B^{p}. Indeed, a cochain c^{p} ∈ S^{p}(X, ∑_{i} A_{i}; R) is a coboudnary in S·(X, ∑_{i} A_{i}; R) if it is of the form δd for some d in S^{p−1}(X, ∑_{i} A_{i}; R). Remark 4.1. It’s easily seen that given A_{1}, …, A_{n}, B_{1}, …, B_{m} ⊂ X, the cup product restricts to a map: (4.3)

∪: S^{p} (X, ∑_{i=1}^{n} A_{i}; R) × S^{q} (X, ∑_{i=1}^{m} B_{i}; R) ⟶ S^{p+q} (X, ∑_{i=1}^{n} A_{i} + ∑_{i=1}^{m} B_{i}; R) .

Obviously, the cup product of cocycles is again a cocycle. Moreover, formula (4.2) shows again that if one of the factors in (4.3) is a coboundary then the cup product is a coboundary, i.e., if c^{p}, c^{q} are both cocycles and if either c^{p} ∈ B^{p}(X, ∑_{i} A_{i}; R) or c^{q} ∈ B^{q}(X, ∑_{i} B_{i}; R) then c^{p} ∪ c^{q} ∈ B^{p+q}(X, ∑_{i} A_{i} + ∑_{i} B_{i}; R). Therefore (4.3) also induces a cup procuct in cohomology.

Our objective is to define a cup product:

(4.4)  ∪: H^{p}(X, A; R) × H^{q}(X, B; R) ⟶ H^{p+q}(X, A ∪ B; R).

This will only be possible if (A, B) is an excisive couple.

Given subsets A, B of a topological space X, we say that (A, B) is an excisive couple if inclusion S(A) + S(B) → S(A ∪ B) induces a homology isomorphism. We have an alternative caracterization for excisive couples.

Proposition 4.2. Given A, B ⊂ X, we have that (A, B) is an excisive couple if and only if inclusion (A, A ∩ B) → (A ∪ B, B) induces a homology isomorphism.

Proof. Consider the following commutative diagram of chain complexes and chain maps:

(4.5) \begin{aligned}0 & ⟶ & S(B) & ⟶ & S(A ∪ B) & ⟶ & S(A ∪ B, B) & ⟶ & 0 \\ & & ↑ & & ↑ & & ↑ \\ 0 & ⟶ & S(B) & ⟶ & S(A) + S(B) & ⟶ & S(A, A ∩ B) & ⟶ & 0 \end{aligned}

where all the arrows are induced by inclusions. The lines are short exact sequences. The left vertical arrow (obviously) induces a homology isomorphism. Considering the corresponding diagram whose lines are long exact homology sequences, we conclude, by applying the five Lemma, that the middle arrow in (4.5) induces a homology isomorphism if and only if the right arrow in (4.5) induces a homology isomorphism. □

The next Lemma allows us to define cup products for relative homology.

Lemma 4.3. Let (A, B) be an excisive couple with A, B ⊂ X. Then the chain map i: S(X, A + B) → S(X, A ∪ B) induced by inclusion induces both a homology isomorphism i_{∗} and a cohomology isomorphism i^{∗}.

Proof. Consider the following commutative diagram of chain complexes and chain maps:

(4.6) \begin{aligned}0 & ⟶ & S(A) + S(B) & ⟶ & S(X) & ⟶ & S(X, A + B) & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & S(A ∪ B) & ⟶ & S(X) & ⟶ & S(X, A ∪ B) & ⟶ & 0 \end{aligned}

where all the arrows are induced by inclusions. The lines are short exact sequences. The left and middle vertical arrows induce homology isomorphisms. Considering the corresponding diagram whose lines are long exact homology sequences, we conclude, by applying the five Lemma, that the right arrow in (4.6) induces a homology isomorphism as well.

Finally, recall that if a chain map between free chain complexes induce a homology isomorphism then it also induces a cohomology isomorphism (see [1, Theorem 45.5]). Since S(X, A + B) and S(X, A ∪ B) are free, the proof is complete. □

Now, if (A, B) is an excisive couple and i is the map in the statement of Lemma 4.3, we define the cup product of (4.4) by:

{c^{p}} ∪ {c^{q}} = (i^{∗})^{−1}({c^{p} ∪ c^{q}}),

for any c^{p} ∈ Z^{p}(X, A), c^{q} ∈ Z^{q}(X, B), where the brackets {·} denote cohomology classes. We want now to prove that, under suitable conditions, the relative cup product (4.4) is associative.

We start with a few lemmas. Lemma 4.4. Let C be a chain complex and let C_{1}, D_{1}, C_{2}, D_{2} be subcomplexes of C with C_{1} ⊂ D_{1} and C_{2} ⊂ D_{2}. Suppose that both inclusions C_{1} → D_{1} and C_{2} → D_{2} induce homology isomorphisms. Then inclusion C_{1} + C_{2} → D_{1} + D_{2} induces a homology isomorphism if and only if inclusion C_{1} ∩ C_{2} → D_{1} ∩ D_{2} induces a homology isomorphism.

Proof. Consider the following commutative diagram of chain complexes and chain maps:

(4.7)  \begin{aligned}0 & ⟶ & C_{1} ∩ C_{2} & ⟶^{j} & C_{1} ⊕ C_{2} & ⟶^{k} & C_{1} + C_{2} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & D_{1} ∩ D_{2} & ⟶_{j} & D_{1} ⊕ D_{2} & ⟶_{k} & D_{1} + D_{2} & ⟶ & 0 \end{aligned}

where the vertical arrows are inclusions, the maps j are of the form j(x) = (x, −x) and the maps k are of the form k(x, y) = x+y. Then the lines in (4.7) are short exact sequences and the middle vertical arrow induces a homology isomorphism. As usual, by considering the long exact homology sequences and applying the five Lemma, we conclude that the left vertical arrow in (4.7) induces a homology isomorphism if and only if the right vertical arrow in (4.7) induces a homology isomorphism. □

Lemma 4.5. Let A, B, C ⊂ X. Suppose that the couples (A, B), (A∪B, C), (B, C) and (A, B ∪ C) are all excisive. Then the following conditions are equivalent:

(1) Inclusion S(A) + S(B) + S(C) → S(A ∪ B ∪ C) induces a homology isomorphism.

(2) The map induced by inclusion S(X, A + B + C) → S(X, A ∪ B ∪ C) induces both homology and cohomology isomorphisms.

(3) (A ∩ C, B ∩ C) is an excisive couple.

(4) (A ∩ B, A ∩ C) is an excisive couple.

Proof. The equivalence between (1) and (2) is proven as in Lemma 4.3. We prove that (1) and (3) are equivalent. The proof that (1) and (4) are equivalent is analogous, simply exchanging the roles of A and C.

Since (A ∪ B, C) is excisive, we see that (1) is equivalent to the condition that inclusion S(A)+S(B)+S(C) → S(A∪B)+S(C) induces a homology isomorphism. Now apply Lemma 4.4 to the complexes C = S(X), C_{1} = S(A) + S(B), D_{1} = S(A ∪ B) and C_{2} = D_{2} = S(C). The conclusion follows by observing that:

\begin{matrix}C_{1} ∩ C_{2} = (S(A) + S(B)) ∩ S(C) = S(A ∩ C) + S(B ∩ C) and \\ D_{1} ∩ D_{2} = S(A ∪ B) ∩ S(C) = S((A ∩ C) ∪ (B ∩ C)). \end{matrix}

□

We can now prove our main theorem. Theorem 4.6. Let A, B, C ⊂ X. Suppose that the couples (A, B), (A ∪ B, C), (B, C) and (A, B ∪ C) are all excisive. Suppose also that one (or all) of the equivalent conditions in the statement of Lemma 4.5 are satisfied. Then for α ∈ H^{p}(X, A; R), β ∈ H^{q}(X, B; R) and γ ∈ H^{r}(X, C; R) we have:

(α ∪ β) ∪ γ = α ∪ (β ∪ γ) ∈ H^{p+q+r}(X, A ∪ B ∪ C; R),

where the cup products are as in (4.4), so that α ∪ β ∈ H^{p+q}(X, A ∪ B; R) and β ∪ γ ∈ H^{q+r}(X, B ∪ C; R).

Proof. Let c^{p} ∈ Z^{p}(X, A; R), c^{q} ∈ Z^{q}(X, B; R) and c^{r} ∈ Z^{r}(X, C; R) be representants of the cohomology classes α, β and γ respectively. Consider the maps

S(X, A + B) ⟶^{i} S(X, A ∪ B), S(X, C + D) ⟶^{j} S(X, C ∪ D)

S(X, A + B + C) ⟶^{m} S(X, (A ∪ B) + C) ⟶^{k} S(X, A ∪ B ∪ C)

S(X, A + B + C) ⟶^{n} S(X, A + (B ∪ C)) ⟶^{l} S(X, A ∪ B ∪ C)

S(X, A + B + C) ⟶^{r} S(X, A ∪ B ∪ C)

induced by inclusions. We have r = k ∘ m = l ∘ n.

By Lemmas 4.3 and 4.5, the maps i, j, k, l and r induce cohomology isomorphisms. Therefore m and n also induce cohomology isomorphisms. Let d ∈ Z^{p+q}(X, A ∪ B; R) and e ∈ Z^{q+r}(X, B ∪ C; R) be representants of the cohomology classes α ∪ β and β ∪ γ respectively. Then:

(4.8)  i^{∗}{d} = {c^{p} ∪ c^{q}} and j^{∗}{e} = {c^{q} ∪ c^{r}},

where the brackets {·} represent cohomology classes. Moreover:

(4.9)  (α ∪ β) ∪ γ = (k^{∗})^{−1}{d ∪ c^{r}} and α ∪ (β ∪ γ) = (l^{∗})^{−1}{c^{p} ∪ e}.

Aplying r^{∗} to both equalities in (4.9) we get:

(4.10)  r^{∗}((α ∪ β) ∪ γ) = m^{∗}{d ∪ c^{r}} and r^{∗}(α ∪ (β ∪ γ)) = n^{∗}{c^{p} ∪ e}.

By (4.8), d and c^{p} ∪ c^{q} define the same cohomology class in S·(X, A + B) and also e and c^{q} ∪ c^{r} define the same cohomology class in S·(X, B + C). Since c^{p}, c^{r} are cocycles, by Remark 4.1 and formula (4.10), we get:

r^{∗}((α ∪ β) ∪ γ) = {c^{p} ∪ c^{q} ∪ c^{r}} = r^{∗}(α ∪ (β ∪ γ)).

Since r^{∗} is an isomorphism, the proof is complete. □

References

[1] J. R. Munkres, Elements of Algebraic Topology. The Benjamin/Cummings Publishing Company, Inc., 1984.

Departamento de Matemática, Universidade de São Paulo, Brazil

   E-mail address: tausk@ime.usp.br
   URL: http://www.ime.usp.br/~tausk