COHOMOLOGY OF GROUPS : EXERCISES

Sheet 1, 4 April 2007

Prerequisites on simplicial sets (and the solution to some of the exercises below!) can be found in Waldhausen’s notes “Algebraische Topologie”, chapter 9, available

at http://www.math.uni-bonn.de/people/schwede/Lehre/Lehre-index.html .

Definition. Let C be a small category, i.e., the class of objects of C is a set. Define the nerve of C to be the simplicial set NC corepresented by Δ in the category of small categories and functors. More explicitly, recall that Δ is the category with set of objects {[n] | n ∈ ℤ, n ⩾ 0}. Here [n] is the ordered set {0, 1, …, n} viewed as a category (we have exactly one morphism k → ℓ if k ⩽ ℓ, and no morphism otherwise). The morphisms from [n] to [m] in Δ are the functors. The nerve of C is given by the contravariant functor

NC : Δ → Sets
[n] ↦ Functors([n], C) .

The geometric realization |NC| of NC is called the classifying space of C, and is denoted BC.

Exercise 1.1. If C is small category and n ⩾ 0 is an integer, give an explicit description of an n-simplex in N_{n}C. Using this description, express the face map d_{i} : N_{n}C → N_{n−1}C and the degeneracy map s_{i} : N_{n}C → N_{n+1}C, for 0 ⩽ i ⩽ n.

Exercise 1.2. Let [n] ∈ Δ be viewed as a small category, as mentioned above. Prove that B[n] is homeomorphic to the standard n-simplex ∇^{n} ⊂ ℝ^{n+1}.

Exercise 1.3. Let C and D be small categories, F and G be functors C → D, and let η : F → G be a natural transformation. Show that BF and BG are freely homotopic as maps BC → BD.

Hint: construct the homotopy H : ∇_{1} × BC → BD between BF and BG as the realization of the nerve of a functor [1] × C → D.

Exercise 1.4. Let G be a group and X be a connected simplicial object in free G-sets, i.e. a contravariant functor X : Δ → G−Sets such for all n ⩾ 0 the G-action on X_{n} is free. Prove that the natural map |X|/G → |X/G| is a homeomorphism. Prove that G acts properly on |X|, i.e., every x ∈ |X| has a neighborhood V such that V ∩ gV = ∅ for all g ∈ G − {1}. Deduce that |X| → |X/G| is a covering with group of deck transformations isomorphic to G.

Typeset by AMS-TEX