Topological Groups 7. Exercise Sheet

Topological Groups 7. Exercise Sheet

Department of Mathematics Summer Term 2013

Andreas Mars 8.7.2013

Groupwork

Exercise G1 (Projective Limits)

LetP ={f_jk:G_k →G_j}be a projective system of topological groups with limit maps f_j: lim_←G_j=G→G_j. Acone overP is a topological groupC together with morphismsγj:C →G_j of topological groups, such that for j ≤kthe identityγj=f_jk◦γkholds. In other words, the following diagram commutes:

C

γ

γj

γk

G=lim_←G_j

f_j

zz ^fk $$

. . .oo G_j G_k

fjk

oo . . .oo

(a) Show that (or rather: Convince yourself that . . . ): The groupGwith the limit mapsf_j:G→G_jis a cone overP. (b) Show that withC:={1}and the obvious mapsγjwe obtain a cone overP.

(c) Prove the following universal property of the projective limit: If{γj:C→G_j}is a cone overP, then there exists a unique morphismγ:C→G=lim_←G_jsuch thatγj=f_j◦γ.

Exercise G2 (Compact Lie Groups) Show that:

(a) Every finite discrete group is a compact Lie group.

(b) A finite direct product of compact Lie groups is a compact Lie group. What about infinite direct products?

(c) A closed subgroup of a compact Lie group is a compact Lie group.

Exercise G3 (Divisibility)

LetAbe an abelian group. Show that the following are equivalent:

(a) Ais divisible.

(b) For everya∈Athere exists a homomorphism f:Q→Asuch that f(1) =a.

1