• Aucun résultat trouvé

COHOMOLOGY OF GROUPS : EXERCISES

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "COHOMOLOGY OF GROUPS : EXERCISES"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

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 N C corepresented by ∆ in the category of small categories and functors. More explicitly, recall that ∆ 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 → ` if k 6 `, and no morphism otherwise). The morphisms from [n] to [m] in ∆ are the functors. The nerve of C is given by the contravariant functor

N C : ∆ → Sets

[n] 7→ Functors([n], C) .

The geometric realization |N C | of N C 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 6 i 6 n.

Exercise 1.2. Let [n] ∈ ∆ 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 : ∆ → 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 byAMS-TEX

1

Références

Télécharger maintenant ( PDF - 1 Page - 69.57 KB )

Documents relatifs

Basic Concepts - Exercises (Chapter 0)

Find the ”stiffness” matrix K for piecewise linear

Exercises - Chapter 0 (correction)

Find the ”stiffness” matrix K for linear basis functions... Therefore the problem is

The Modal Auxiliaries : Exercises

b) Je ne sais pas parler anglais, j'habite Londres depuis trois semaines.. c) Demain, je ne pourrai venir à l'école. La stib fait

The Modal Auxiliaries : Exercises

i) A-t-il déjà dû aller au Brésil depuis qu'il a commencé à travailler pour nous ? j) L'année prochaine, tu ne pourras plus aller au lit si tard. Papa ne

Exercises for Chapter 3

Let us see here some illustrations of the algorithm (seen in course) which gives the factorization of a polynomial over a number field.. For this, if you are lazy or tired, you

Exercises for Chapter 4

These numbers appear also in Gauss theorem on the constructibility (with rule and compas) of regular polygones: a regular polygone with n vertices is constructible if and only if n is

Exercises for Chapter 5

Compute the class group, the regulator and a system of fundamental units for the number fields defined by the

Solutions of Exercises

The most suitable one in this case claims that unitary operators are reversible and save the unit length.. Thanks to the superposition principle it is enough to check these