Homologie et Cohomologie, April 20, 2007, Exercises 5
1 Simplicial models for S
2Explain why∂∆ [3] and ∆ [2]/∂∆ [2] are both ”simplicial models” forS2.
2 Kan complexes
Definition 2.1 A simplicial set K• is a Kan complex if ∀f ∈sSet(Λk[n], K•),
∃fˆ∈sSet(∆ [n], K), such that
Λk[n] _ f //
K
∆ [n]
fˆ
>>
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commutes.
Explicitly, this means that ∀ {x0, ..., xk−1, xk+1, ..., xn} ⊂ Kn−1, such that dixj = dj−1xi ∀i < j, i, j6=k, ∃y∈Kn, such that diy=xi ∀i6=k.
Show that the simplicial set S•(X) of any topological spaceX is a Kan complex.
1
3 Mapping spaces (harder)
Define a simplicial mapping spaceM ap(K•, L•)as follows:
M ap(K•, L•)n=sSet(K•×∆ [n], L•).
1. Give explicit formulas for the faces and degeneracies and explain why the simplicial identities hold.
2. Define a composition morphism
M ap(K•, L•)×M ap(L•, M•)−→M ap(K•, M•) extending
sSet(K•, L•)×sSet(L•, M•)−→sSet(K•, M•).
2
