The Cone Operator in Singular Homology

This note reorganizes some material in Hatcher’s “Algebraic Topology”, §2.1.

We introduce the cone operator, as a tool for the homotopy and excision axioms.

Linear chains As a first step, we study the following situation. Suppose Z is a convex subspace of ℝ^{q}. Given any n + 1 points v_{0}, v_{1}, …, v_{n} in Z (not necessarily linearly independent), we have the linear map

λ = [v_{0}, v_{1}, …, v_{n}]: Δ^{n} − ; − ; → Z  (1)

from the standard n-simplex Δ^{n} ⊂ ℝ^{n+1} with vertices e_{i} (for 0 ≤ i ≤ n), defined by λ(e_{i}) = v_{i} for each i. Its image in Z is thus

λ(Δ^{n}) = {∑_{i=0}^{n} t_{i}v_{i} : t_{i} ≥ 0 for all i, ∑_{i=0}^{n} t_{i} = 1} ,  (2)

which may or may not be a geometric simplex.

Such maps λ generate the subgroup LC_{n}(Z) ⊂ C_{n}(Z) of linear n-chains in Z. As n varies, the groups LC_{n}(Z) clearly form a sub-chain complex LC_{∗}(Z) of the singular chain complex C_{∗}(Z). The space Z, being convex, is contractible.

Proposition 3 H_{0}(LC_{∗}(Z)) ≅ ℤ, and H_{n}(LC_{∗}(Z)) = 0 for n ≠ 0.

Suppose Z′ ⊂ ℝ^{s} is another convex subspace, and that A: ℝ^{q} → ℝ^{s} is a linear (or affine) map that satisfies A(Z) ⊂ Z′. It is clear that the chain map A_{#}: C_{∗}(Z) → C_{∗}(Z′) restricts to a chain map

A_{#}: LC_{∗}(Z) − ; − ; → LC_{∗}(Z′).  (4)

The cone operator This is suggested by the fact that the cone on a simplex is another simplex.

Definition 5 Given a point v ∈ Z, we define the cone operator

C_{v}: LC_{n}(Z) − ; − ; → LC_{n+1}(Z)

as the homomorphism defined on each linear simplex (1) by

C_{v}λ = C_{v}[v_{0}, v_{1}, v_{2}, …, v_{n}] = [v, v_{0}, v_{1}, v_{2}, …, v_{n}] ∈ LC_{n+1}(Z).  (6)

Proposition 7 With A: Z → Z′ as in (4), we have

A_{#} ^{∘} C_{v} = C_{Av} ^{∘} A_{#}: LC_{∗}(Z) − ; − ; → LC_{∗}(Z′).

Let us compute the boundary.

\begin{aligned}∂C_{v}λ & = [v_{0}, v_{1}, …, v_{n}] − ∑_{i=0}^{n}(−1)^{i}[v, v_{0}, v_{1}, …, v̂_{i}, …, v_{n}] \\ & = λ − C_{v}∂λ. \end{aligned}

This calculation is valid only for n > 0. For λ = [v_{0}] we have instead

∂C_{v}λ = ∂[v, v_{0}] = [v_{0}] − [v] = λ − [v].

By taking linear combinations, for any chain c ∈ LC_{n}(Z) we may write

∂C_{v}c = \{ \begin{matrix}c − C_{v}∂c & if n > 0; \\ c − (ϵc)[v] & if n = 0; \end{matrix}  (8)

where we introduce the augmentation homomorphism ϵ: LC_{0}(Z) = C_{0}(Z) → ℤ given by ϵ[z] = 1 for all z ∈ Z.

Proof of Proposition 3 It is not necessary to determine the n-cycles or n-boundaries in LC_{∗}(Z). If c ∈ LC_{n}(Z) is a cycle, where n > 0, (8) shows that c = ∂C_{v}c and is thus a boundary.

Any chain c ∈ LC_{0}(Z) is a 0-cycle, and by (8) is homologous to some multiple (ϵc)[v] of [v]. It follows that H_{0}(LC_{∗}(Z)) ≅ ℤ, generated by [v].

Chain homotopies The usefulness of equation (8) suggests a definition.

Definition 9 Given two chain maps f, g: C → C′ between chain complexes C and C′, a chain homotopy from f to g is a family of homomorphisms s_{n}: C_{n} → C′_{n+1} that satisfy

∂s_{n}c + s_{n−1}∂c = g_{n}c − f_{n}c for all n and all c ∈ C_{n}.  (10)

We say the two chain maps are chain homotopic and write f ≃ g: C → C′.

It is easy to see that being chain homotopic is an equivalence relation. The argument of Proposition 3 generalizes immediately.

Proposition 11 If f, g: C → C′ are chain homotopic chain maps, we have f_{∗} = g_{∗}: H_{n}(C) → H_{n}(C′).

Proof If c ∈ C_{n} is a cycle, (10) reduces to g_{n}c − f_{n}c = ∂s_{n}c, which shows that the cycles f_{n}c and g_{n}c are homologous.

The prism operator, on linear chains Let Z be a convex subspace of ℝ^{q}, as before. We compare the two chain maps j_{0#}, j_{1,#}: LC_{∗}(Z) → LC_{∗}(Z×I) induced by the maps j_{0} and j_{1} defined by j_{r}(z) = (z, r).

Given a linear n-simplex λ = [u_{0}, u_{1}, …, u_{n}] in Z, we write the two simplices j_{0} ^{∘} λ and j_{1} ^{∘} λ as [v_{0}, v_{1}, …, v_{n}] and [w_{0}, w_{1}, …, w_{n}] respectively; they give the two ends of a prism Δ^{n} × I in Z × I (when the u_{i} are linearly independent). Geometrically, although the prism is not a simplex, it can be regarded as a cone with vertex v_{0} on the union of the top face Δ^{n} ×1 and F_{0} ×I, where F_{0} denotes the face of Δ^{n} opposite u_{0} (the image of η_{0}: Δ^{n−1} ⊂ Δ^{n}). (See the pictures on page 112.) This suggests the following algebraic definition.

Definition 12 We define the prism operator homomorphism

P′_{n}: LC_{n}(Z) − ; − ; → LC_{n+1}(Z×I)

on any linear n-simplex λ = [u_{0}, u_{1}, …, u_{n}] as above by

P′_{n}λ = C_{v_{0}}(j_{1#}λ − P′_{n−1}d_{0}λ) ∈ LC_{n+1}(Z×I) for n ≥ 0,  (13)

using induction on n, where v_{0} = (u_{0}, 0) ∈ Z × I. Of course, P′_{n} = 0 for all n < 0.

Note that the vertex v_{0} of the cone varies as λ varies.

It is clear that P′_{n} commutes with linear maps, in the following sense.

Proposition 14 Let A: Z → Z′ be a linear (or affine) map as in (4). Then

P′_{n} ^{∘} A_{#} = (A×1_{I})_{#} ^{∘} P′_{n}: LC_{n}(Z) − ; − ; → LC_{n+1}(Z′×I).

Lemma 15 Let Z be a convex subspace of ℝ^{q}. Then for all n,

∂ ^{∘} P′_{n} + P′_{n−1} ^{∘} ∂ = j_{1#} − j_{0#}: LC_{n}(Z) − ; − ; → LC_{n}(Z×I).  (16)

In words, the homomorphisms P′_{n} form a chain homotopy P′ from j_{0#} to j_{1#}.

Proof For n = 0, P′_{0}λ = [v_{0}, w_{0}] and

∂P′_{0}λ = ∂[v_{0}, w_{0}] = [w_{0}] − [v_{0}] = j_{1#}λ − j_{0#}λ.

For n > 0, we proceed inductively:

∂P′_{n}λ = ∂C_{v_{0}}j_{1#}λ − ∂C_{v_{0}}P′_{n−1}d_{0}λ by (13)
= j_{1#}λ − C_{v_{0}}∂j_{1#}λ − P′_{n−1}d_{0}λ + C_{v_{0}}∂P′_{n−1}d_{0}λ by (8)

= j_{1#}λ − C_{v_{0}}j_{1#}∂λ − P′_{n−1}d_{0}λ − C_{v_{0}}P′_{n−2}∂d_{0}λ + C_{v_{0}}j_{1#}d_{0}λ − C_{v_{0}}j_{0#}d_{0}λ,

using (16) for n − 1 and the induction hypothesis. Here, the second and fifth terms combine as

− ∑_{i=1}^{n}(−1)^{i}C_{v_{0}}j_{1#}d_{i}λ.  (17)

In the fourth term, we expand

∂d_{0}λ = ∑_{k=0}^{n−1}(−1)^{k}d_{k}d_{0}λ = ∑_{k=0}^{n−1}(−1)^{k}d_{0}d_{k+1}λ,

so that the fourth term becomes

−C_{v_{0}}P′_{n−2}∂d_{0}λ = ∑_{k=0}^{n−1}(−1)^{k+1}C_{v_{0}}P′_{n−2}d_{0}d_{k+1}λ.  (18)

The last term reduces to one we want,

C_{v_{0}}j_{0#}[u_{1}, …, u_{n}] = C_{v_{0}}[v_{1}, …, v_{n}] = [v_{0}, v_{1}, …, v_{n}] = j_{0#}λ.

Meanwhile, again using (13), for P′_{n−1},

\begin{aligned}P′_{n−1}∂λ & = P′_{n−1}d_{0}λ + ∑_{i=1}^{n}(−1)^{i}P′_{n−1}d_{i}λ \\ & = P′_{n−1}d_{0}λ + ∑_{i=1}^{n}(−1)^{i}C_{v_{0}}j_{1#}d_{i}λ − ∑_{i=1}^{n}(−1)^{i}C_{v_{0}}P′_{n−2}d_{0}d_{i}λ, \end{aligned}

where we note that for i > 0, the leading vertex of d_{i}λ is still u_{0}. When we add ∂P′_{n}λ to this, the first term cancels out, the second term cancels (17), and the third cancels (18), if we replace i by k + 1.

The prism operator, on general chains We extend the prism operator P′ to general spaces and chains by working in the space Δ^{n} × I rather than X or Y. We focus attention on the element δ_{n} ∈ LC_{n}(Δ^{n}) ⊂ C_{n}(Δ^{n}), which denotes the identity map 1: Δ^{n} → Δ^{n}, considered as a singular n-simplex of the space Δ^{n}. Then for any singular n-simplex σ: Δ^{n} → X, the chain map σ_{#}: C_{∗}(Δ^{n}) → C_{∗}(X) induces σ_{#}δ_{n} = σ.

Definition 19 Given a homotopy f_{t}: X → Y, hence a map F: X ×I → Y, we define the prism operator homomorphism P_{n}: C_{n}(X) → C_{n+1}(Y) on the singular n-simplex σ: Δ^{n} → X, a generator of C_{n}(X), by

P_{n}σ = F_{#}(σ×1_{I})_{#}P′_{n}δ_{n}.  (20)

Remark We note that if X is a convex subspace of ℝ^{q}, σ a linear singular n-simplex of X, and Y = X × I with F the identity map of X × I, equation (20) reduces to

P_{n}σ = (σ×1_{I})_{#}P′_{n}δ_{n} = P′_{n}σ_{#}δ_{n} = P′_{n}σ,

with the help of Proposition 14. So P_{n} does extend P′_{n}.

Theorem 21 Given a homotopy f_{t}: X → Y, the resulting prism operator satisfies

∂ ^{∘} P_{n} + P_{n−1} ^{∘} ∂ = f_{1#} − f_{0#}: C_{n}(X) − ; − ; → C_{n}(Y).  (22)

Corollary 23 Then f_{∗} = g_{∗}: H_{n}(X) → H_{n}(Y).

Proof of Theorem 21 We evaluate each term of (22) on a general singular n-simplex σ: Δ^{n} → X:

∂P_{n}σ = ∂F_{#}(σ×1_{I})_{#}P′_{n}δ_{n} = F_{#}(σ×1_{I})_{#}∂P′_{n}δ_{n};
P_{n−1}∂σ = ∑_{i=0}^{n}(−1)^{i}P_{n−1}(σ ^{∘} η_{i}) = ∑_{i=0}^{n}(−1)^{i}F_{#}(σ×1_{I})_{#}(η_{i}×1_{I})_{#}P′_{n−1}δ_{n−1}

= F_{#}(σ×1_{I})_{#}P′_{n−1} {∑_{i=0}^{n}(−1)^{i}η_{i#}δ_{n−1}} by Proposition 14

= F_{#}(σ×1_{I})_{#}P′_{n−1}∂δ_{n};

f_{1#}σ = f_{1} ^{∘} σ = F ^{∘} (σ×1_{I}) ^{∘} j_{1} = F_{#}(σ×1_{I})_{#}j_{1#}δ_{n};
f_{0#}σ = F_{#}(σ×1_{I})_{#}j_{0#}δ_{n}, similarly.

Then (22) follows from (16).

Barycentric subdivision, linear case We define the barycentric subdivision first on linear simplices, to produce a chain map S′_{n}: LC_{n}(Z) → LC_{n}(Z). Geometrically, we proceed by induction; once the faces of a linear n-simplex λ as in (1) have been subdivided, we join everything to the barycenter b(λ) = \frac{1}{n+1} ∑_{i=0}^{n} v_{i} of λ.

Algebraically, we begin the induction with S′_{0} = 1, and continue with

S′_{n}λ = C_{b(λ)}S′_{n−1}∂λ ∈ LC_{n}(Z) for n > 0.  (24)

(Of course, S′_{n} = 0 for n < 0.) We must check that S′ is a chain map. For n = 1 we have, using (8),

∂S′_{1}λ = ∂C_{b(λ)}S′_{0}∂λ = S′_{0}∂λ − ϵ(S′_{0}∂λ)[b(λ)] = S′_{0}∂λ,

since ϵ(S′_{0}∂λ) = ϵ(∂λ) = 1 − 1 = 0. For n ≥ 2 we compute, again using (8),

∂S′_{n}λ = ∂C_{b(λ)}S′_{n−1}∂λ = S′_{n−1}∂λ − C_{b(λ)}∂S′_{n−1}∂λ = S′_{n−1}∂λ,

since by induction ∂S′_{n−1}∂λ = S′_{n−2}∂∂λ = 0.

We also need a chain homotopy T′ between S′ and the identity chain map 1, i. e. homomorphisms T′_{n}: LC_{n}(Z) → LC_{n+1}(Z) that satisfy

∂ ^{∘} T′_{n} + T′_{n−1} ^{∘} ∂ = 1 − S′_{n}: LC_{n}(Z) − ; − ; → LC_{n}(Z) for all n.  (25)

Geometrically (see the picture on page 122), we subdivide the lower face Δ^{n} × 0 of the prism Δ^{n} × I, leave the upper face Δ^{n} × 1 alone, and join the barycenter (b(λ), 0) of the lower face to Δ^{n} × 1 and the already subdivided vertical faces ∂Δ^{n} × I of the prism, then project everything to Z.

Algebraically, we begin the induction with T′_{0} = 0. (Hatcher uses T′_{0}[v_{0}] = [v_{0}, v_{0}], which works equally well. Again, T′_{n} = 0 for n < 0.) We continue with

T′_{n}λ = C_{b(λ)}(λ − T′_{n−1}∂λ) ∈ LC_{n+1}(Z) for n > 0.  (26)

Then equation (25) is trivial for n = 0. We verify it for n > 0 by induction, by evaluating it on λ, with the help of (8),

∂T′_{n}λ = ∂C_{b(λ)}(λ − T′_{n−1}∂λ) = λ − T′_{n−1}∂λ − C_{b(λ)}∂λ + C_{b(λ)}∂T′_{n−1}∂λ.

We use (25) for n − 1 to expand the last term,

C_{b(λ)}∂T′_{n−1}∂λ = −C_{b(λ)}T′_{n−2}∂∂λ + C_{b(λ)}∂λ − C_{b(λ)}S′_{n−1}∂λ.

The first term on the right vanishes, the second cancels the unwanted term in ∂T′_{n}λ, and the third is S′_{n}λ by definition.

The following property of S′_{n} and T′_{n} is immediate.

Proposition 27 With A: Z → Z′ as in (4), the chain map A_{#}: LC_{n}(Z) → LC_{n}(Z′) commutes with S′_{n} and T′_{n}: S′_{n} ^{∘} A_{#} = A_{#} ^{∘} S′_{n} and T′_{n} ^{∘} A_{#} = A_{#} ^{∘} T′_{n}.

Barycentric subdivision, general case As with the prism operator, we extend the definition of S′_{n} and T′_{n} to a general singular n-simplex σ: Δ^{n} → X of any space X by using the chain map σ_{#}: LC_{n}(Δ^{n}) ⊂ C_{n}(Δ^{n}) → C_{n}(X), where δ_{n} ∈ LC_{n}(Δ^{n}) denotes as before the singular simplex 1: Δ^{n} → Δ^{n}. For all n, we define the homomorphisms S_{n}: C_{n}(X) → C_{n}(X) and T_{n}: C_{n}(X) → C_{n+1}(X) on each generator σ by

S_{n}σ = σ_{#}S′_{n}δ_{n};  T_{n}σ = σ_{#}T′_{n}δ_{n}.  (28)

If X happens to be a convex subspace of a real vector space and σ is a linear (or affine) n-simplex, these reduce by Proposition 27 to S′_{n}σ and T′_{n}σ.

To verify that S is a chain map, we expand, using the fact that σ_{#} and S′ are chain maps,

\begin{aligned}∂S_{n}σ & = ∂σ_{#}S′_{n}δ_{n} = σ_{#}∂S′_{n}δ_{n} = σ_{#}S′_{n−1}∂δ_{n}, \\ S_{n−1}∂σ & = ∑_{i=0}^{n}(−1)^{i}S_{n−1}(σ ^{∘} η_{i}) = ∑_{i=0}^{n}(−1)^{i}σ_{#}η_{i#}S′_{n−1}δ_{n−1}. \end{aligned}

These agree, since

∂δ_{n} = ∑_{i=0}^{n}(−1)^{i}δ_{n} ^{∘} η_{i} = ∑_{i=0}^{n}(−1)^{i}η_{i} = ∑_{i=0}^{n}(−1)^{i}η_{i#}δ_{n−1}  (29)

and η_{i#} commutes with S′ by Proposition 27.

A similar proof shows that T satisfies

∂T_{n}σ + T_{n−1}∂σ = σ − S_{n}σ: C_{n}(X) − ; − ; → C_{n}(X),  (30)

so is a chain homotopy from S to 1. We expand each term,

\begin{aligned}∂T_{n}σ & = ∂σ_{#}T′_{n}δ_{n} = σ_{#}∂T′_{n}δ_{n} = σ_{#}T′_{n}∂δ_{n}, \\ T_{n−1}∂σ & = ∑_{i=0}^{n}(−1)^{i}T_{n−1}(σ ^{∘} η_{i}) = ∑_{i=0}^{n}(−1)^{i}σ_{#}η_{i#}T′_{n−1}δ_{n−1}, \\ σ & = σ_{#}δ_{n}, \\ S_{n}σ & = σ_{#}S′_{n}δ_{n}. \end{aligned}

Since T′_{n−1} commutes with η_{i#} by Proposition 27, (30) now follows from (25).

We also need to know that S′_{n} subdivides any linear simplex λ as in (1) into simplices that really are smaller. There are (n+1)! simplices in the barycentric subdivision of λ; one of these is

λ′ = [v_{0}, \frac{v_{0} + v_{1}}{2} , \frac{v_{0} + v_{1} + v_{2}}{3} , …, \frac{v_{0} + v_{1} + … + v_{n}}{n + 1} ] .

All the others are obtained by reordering the vertices of λ. In terms of equation (2), its image in Z ⊂ ℝ^{q} is

λ′(Δ^{n}) = {∑_{i=0}^{n} t_{i}v_{i} : t_{0} ≥ t_{1} ≥ … ≥ t_{n} ≥ 0, ∑_{i=0}^{n} t_{i} = 1}  (31)

Because t_{0} is the largest and ∑_{i=0}^{n} t_{i} = 1, we must have t_{0} ≥ \frac{1}{n+1}.

Lemma 32 We have diam λ′(Δ^{n}) ≤ \frac{n}{n+1}diam λ(Δ^{n}).

Proof Consider the linear simplex λ′′ = [v_{0}, v′′_{1}, v′′_{2}, …, v′′_{n}], where v′′_{i} = \frac{1}{n+1}v_{0}+ \frac{n}{n+1}v_{i} for 1 ≤ i ≤ n. This simplex is geometrically similar to λ, except shrunk in all directions by the factor \frac{n}{n+1}, keeping v_{0} fixed; therefore diam λ′′(Δ^{n}) = \frac{n}{n+1}diam λ(Δ^{n}).

Now the image λ′′(Δ^{n}) consists of all the points of λ(Δ^{n}) with t_{0} ≥ \frac{1}{n+1}, and therefore contains λ′(Δ^{n}).