Topological Groups 1. Exercise Sheet
Department of Mathematics Summer Term 2013
Andreas Mars 17.4.2013
Groupwork
Exercise G1 (Homogeneous Spaces) Give a proof or a counterexample:
(a) A discrete or indiscrete topological space is homogeneous.
(b) The open interval(0,∞)⊆Rwith the order topology is homogeneous.
(c) The compact unit interval[0, 1]⊆Rwith the order topology is homogeneous.
Exercise G2 (Topological Groups)
(a) Prove: The matrix groupsGLn(R), GLn(C), SLn(R)andSLn(C)equipped with the subspace topology induced from Rn×nresp.Cn×nare topological groups.
(b) Show that all groups from part (a) are closed subgroups ofGLn(C). Is one of them an open subgroup?
(c) Show that forF∈ {R,C}the groupGLn(F)is open inFn×n, whileSLn(F)is closed inFn×n. (d) Prove: A groupGequipped with a topologyτis a topological group if and only if the map
G×G→G (g,h)7→gh−1
is continuous with respect toτ.
(e) LetH≤Gbe an open subgroup. Give a proof or counterexample: ThenHis a closed subgroup.
Exercise G3 (Subgroups and Quotients) Show:
(a) LetH≤Gbe a subgroup of a topological groupG. Then the map (g,g0H)7→g g0H
in continuous in the second argument.
(b) IfG/His Hausdorff, then His a closed subgroup. Give necessary and sufficient condition(s) on Hsuch that the quotientG/His a topological group again.
Exercise G4 (Topological Manifolds)
Prove Proposition 1.6: A connected Hausdorff topological manifoldX is homogeneous.
(a) Show that for every two pointsx,yin the open unit ballB1(0)ofRnthere exists a homeomorphism of the closed unit ballB1(0)which mapsxto y and which fixes the boundary∂B1(0)pointwise.
(b) For x∈X andU an open neighbourhood ofx, there exists an open neighbourhoodV of xwith the property that V ⊆Uand the property that for allv∈V there is a homeomorphism f:V →V with f(x) =v, which extends to a homeomorphism ofX. Hint: Where can you use (i)?
(c) For arbitraryx,y∈X there exists a compact line segment connectingx and y. Hint: Now you’re done, using (ii).
Why?
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Homework
Exercise H1 (More Topological Groups) Prove:
(a) The torusT:={z∈C| |z|=1}equipped with usual complex multiplication and the subspace topology fromCis a topological group.
(b) The spaceR/Zwith the quotient topology and the operation(x,y)7→x+y(mod 1)is a topological group.
(c) The exponential mapexp:R/Z→T,x7→e2πi x is a group homomorphism which is bijective, continuous and open (i.e. an isomorphism of topological groups).
Exercise H2 (Construction of Topological Groups) Prove Proposition 2.6:
(a) Let G be a topological group and let H ≤ Gbe a subgroup equipped with the subspace topology. Then H is a topological group.
(b) Let{Gi}i∈I be an arbitrary family of topological groups and equip the cartesian productQ
i∈IGi with the usual product topology. ThenQ
i∈IGiis a topological group.
(c) LetN ÃGbe a normal subgroup and equipG/N with the quotient topology. ThenG/N is a topological group.
Also try to show thatG/Nis Hausdorff if and only ifN=N. Exercise H3 (A Quotient)
Consider the additive groupRwith the usual order topology and its subgroupQ. SinceRis abelian, every subgroup is normal, hence in particularQis and we may consider the quotient groupG:=R/Q.
(a) Remind yourself thatGis a topological group. Is it Hausdorff, T1and/orT0? (b) Prove or disprove: The subgroup{1}is dense inG.
Hint: Compare the groupsG/{1}andR/Q. Do you notice something?
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