472 APPENDIX A. HOMOLOGICAL METHODS∗

f#∂̂ ([v0, · · · ,vn]) ≡ f#

(n

∑k=0

(−1)k [v0, · · · , v̂k, · · · ,vn]

)

≡n

∑k=0

(−1)k[

f (v0) , · · · , f̂ (vk), · · · , f (vn)]

and these are the same. As to the claim about the barycenter,

f (b) = f

(n

∑k=0

1n+1

vk

)=

n

∑k=0

1n+1

f (vk)

which is the barycenter of [ f (v0) , · · · , f (vn)] = f ([v0, · · · ,vn]). Then note that, as dis-cussed earlier, one obtains a result on An (C) with ∂ in place of ∂̂ . ■

I will use Ŝ as the subdivision operator for either C̃ or C. Also I will use T̂ as de-scribed above for either C or C̃. Now if f is an affine map f : C→ C̃ where C,C̃ are convexspaces, it is also the case that f# commutes with T̂ and with Ŝ . Consider first Ŝ . Letc ∈ Gn (C) be a chain in Gn (C). Then the geometric simplices corresponding to Ŝ (c) are∑

ni=0 (−1)i [b,u0, · · · ,0, · · ·un] where b is the barycenter. Then the geometric simplices

corresponding to f#

(Ŝ (c)

)would be ∑

ni=0 (−1)i [ f (b) , f (u0) , · · · ,0, · · · , f (un)] Now

consider Ŝ ( f# (c)) = Ŝ ( f ◦ c) . The geometric simplices associated with this would in-volve the barycenter of f (u0) , ..., f (un) which is 1

n+1 ∑i f (ui) = f( 1

n+1 ∑iui)= f (b) .

Thus the geometric simplices associated will be the same as the above. Hence f#Ŝ = Ŝ f#if f is affine. That f# commutes with T̂ is similar and follows from the observation thatsince f is affine, f of a barycenter of some vectors equals the barycenter of f of thesevectors in the same way as just noted. Now, as noted above, this leads to the same resultsfor An (C) the chains of affine singular simplices. The following is a summary.

Lemma A.5.14 If f : C→ C̃ convex sets, f affine, the following hold:

f# (An (C)) ⊆ An(C̃), f#T̂ = T̂ f#, f#Ŝ = Ŝ f#,

Ŝ ∂̂ = ∂̂Ŝ , ∂̂ T̂ + T̂ ∂̂ = Ŝ − id

Next is a specific example of an affine map from σ p to σ p.

Definition A.5.15 Do the subdivision operator Ŝ on [v0, · · · ,vp] multiple times to obtaina small simplices {σ k}N

k=1 contained in σ p. Then for one of these σ k =[wk

0, · · · ,wkp],

define a map πk : σ p→ σ k ⊆ σ p by πk(∑

pi=0 tivi

)≡ ∑

pi=0 tiwk

i where ti ≥ 0,∑i ti = 1. Theorder of the vertices in σ k is determined by the construction in the subdivision operator.Since πk is affine, we can regard πk as a homomorphism on An (σ p).

The above partition operator allows consideration of singular simplices of the formφ ◦πk in which φ is a singular simplex, a continuous mapping defined on σ p having valuesin some topological space X , not necessarily convex. These compositions will have “small”image. It is important to use something like these to identify homology groups. Onewonders whether φ ∈ Sn (X) is homologous to φ # ∑k πk. Note that τn, the identity map onσn is obviously an affine map.