Barycentric Subdivision

This note parallels material on pages 119–123 of Hatcher.

Linear simplices Let Y be a convex subspace of ℝ^{q}. Given points v_{0}, v_{1}, …, v_{n} in Y (not necessarily independent), recall the linear n-simplex

λ = [v_{0}, v_{1}, …, v_{n}]: Δ^{n} ⟶ Y,  (1)

which is the restriction to Δ^{n} of the linear map ℝ^{n+1} → ℝ^{q} that sends the standard basis vector e_{i} to v_{i} for each i. The barycenter or centroid of this linear simplex is the point b(λ) = ∑_{i} v_{i}/(n+1). The linear simplices generate the group LC_{n}(Y) of linear chains on Y.

Cone operator Given a point y ∈ Y, we define the cone operator homomorphism C_{y}: LC_{n}(Y) → LC_{n+1}(Y) by

C_{y}λ = C_{y}[v_{0}, v_{1}, …, v_{n}] = [y, v_{0}, v_{1}, …, v_{n}].  (2)

Geometrically, we join everything to the point y.

We compute

∂C_{y}λ = [v_{0}, v_{1}, …, v_{n}] − ∑_{i=0}^{n}[y, v_{0}, …, v̂_{i}, …, v_{n}] = λ − C_{y}∂λ,

assuming that n > 0. If n = 0, this calculation is not valid (unless one introduces the empty (-1)-simplex [ ] as in Hatcher); instead we find ∂C_{y}[v_{0}] = ∂[y, v_{0}] = [v_{0}] − [y]. We combine these as

∂C_{y}λ + C_{y}∂λ = λ − r_{y}λ in LC_{n}(Y)  (3)

for all n, where the chain map r_{y}: LC(Y) → LC(Y) is given by r_{y}λ = 0 for n > 0 and r_{y}[v_{0}] = [y]. Thus C_{y} is a contracting chain homotopy for LC(Y), which expresses algebraically the contraction of the convex subspace Y to the point y.

Barycentric subdivision We define the barycentric subdivision first on linear simplices, to produce a chain map S_{n}: LC_{n}(Y) → LC_{n}(Y). Geometrically, we proceed by induction; once the faces of λ have been subdivided, we join everything to the barycenter b(λ) of λ.

Algebraically, we begin the induction with S_{0} = 1, and continue with

S_{n}λ = C_{b(λ)}S_{n−1}∂λ for n > 0.  (4)

(Of course, S_{n} = 0 for n < 0.) The form of this definition, with equation (3), implies that S is a chain map. For n ≥ 2 we compute

∂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. For n = 1, there is an extra term r_{y}S_{0}∂λ = r_{y}∂λ, which vanishes.

Chain homotopy We need a chain homotopy T between S and the identity chain map 1, i. e. T_{n}: LC_{n}(Y) → LC_{n+1}(Y) that satisfies

∂^{∘} T_{n} + T_{n−1} ^{∘} ∂ = 1 − S_{n} for all n.  (5)

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

Algebraically, we begin the induction with T_{0}[v_{0}] = 0. (Of course, T_{n} = 0 for n < 0. Hatcher uses in effect T_{0}[v_{0}] = [v_{0}, v_{0}], but both choices satisfy equation (5) for n = 0.) We continue with

T_{n}λ = C_{b(λ)}(λ − T_{n−1}∂λ) for n > 0.  (6)

We verify equation (5) for n > 0 by using equation (3),

∂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.

By induction, we have, from equation (5) for n − 1,

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 second term on the left vanishes, and the second term on the right is S_{n}λ, by definition.

The following property of S_{n} and T_{n} is immediate.

Lemma 7 Let A: ℝ^{q} → ℝ^{r} be a linear map, and Y′ a convex subspace of ℝ^{r} such that A(Y) ⊂ Y′. Then the chain map A_{#}: LC_{n}(Y) → LC_{n}(Y′) commutes with S_{n} and T_{n}, S_{n} ^{∘} A_{#} = A_{#} ^{∘} S_{n} and T_{n} ^{∘} A_{#} = A_{#} ^{∘} T_{n}.

General singular simplices We extend the definition of S_{n} and T_{n} to a singular nsimplex 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) and noting that σ = σ_{#}[e_{0}, e_{1}, …, e_{n}]. For all n, we define

S_{n}σ = σ_{#}S_{n}[e_{0}, e_{1}, …, e_{n}]  (8)

and

T_{n}σ = σ_{#}T_{n}[e_{0}, e_{1}, …, e_{n}].  (9)

These are consistent with previous definitions by Lemma 7 if X happens to be a real vector space and σ is a linear map.

To verify that S remains a chain map, we compute

∂S_{n}σ = ∂σ_{#}S_{n}[e_{0}, e_{1}, …, e_{n}] = σ_{#}S_{n−1}∂[e_{0}, e_{1}, …, e_{n}],

since σ_{#} and the linear version of S are chain maps. On the other side, equation (8) for the face d_{i}σ = σ^{∘} η_{i} gives

S_{n−1}d_{i}σ = σ_{#}η_{i#}S_{n−1}[e_{0}, e_{1}, …, e_{n−1}].

By Lemma 7, we may rewrite this as

σ_{#}S_{n−1}η_{i#}[e_{0}, e_{1}, …, e_{n−1}] = σ_{#}S_{n−1}[e_{0}, …, ê_{i}, …, e_{n}].

Now we take alternating sums over i.

A similar proof shows that T continues to satisfy equation (5).