MTH3483

|TOPICSINTOPOLOGY|

FALL1998

PROF.A.SUCIU

Homework2

1.Letªbe aprincipalG-bundle.Show:
(a)f^§(ª) isaprincipalG-bundle.
(b)If G acts eÆectively on F , then f ^§ (ª )[F ] ª f ^§ (ª [F ]).

2.GivenaprincipalG-bundleP!Bandaclosed subgroupHofG, show:
(a)P=H!BisaG-bundle withØberG=HassociatedtoP!B.
(b)IfHis alsoanormal subgroup ofG, thenP=H!BisaprincipalG=H-bundle.

3.LetGbe atopological group actingon aspaceX.Show:
(a)IfA is an open subset of X,thenG¢Aisopen.
(b)IfGiscompact andA is a closed subset of X , thenG¢Ais closed.
(b)IfGiscompact andA is a compact subset of X , thenG¢Ais compact.
(d)IfXis HausdorÆ, thenX^GisHausdorÆ.
(e)IfGis compact andXis HausdorÆ, thenX=GisHausdorÆ.
(f)The compactness assumption in part(e)isnecessary.
[Hint:FindanR-action onR²such thatR²=RisnotHausdorÆ.] 4.Letª_nbetheprincipalZ_n-bundlep_n:S¹!S¹,wherep_n(z)=zⁿ.Considerthe
associatedZ_n-bundle¥_n=ª_n[S¹],whereZ_n actsonS¹by left-translation.Show
that¥_nis nottrivial as aZ_n-bundle, butit istrivial as a(principal)S¹-bundle. 5.Consider theZ_n-action onS³givenby (z₁;z₂)7!(ªz₁;ªz₂), whereª=e^2ºi=n.LetL_nbe
the orbitspace.
(a)DeØne aprincipalS¹-bundleL_n!S².
(b)What isthe clutching function ofthisbundle?
(c)ShowthatL₁=S³andL₂= SO(3).

6.Letp:E!Bbe areal vectorbundle of rankk.Let
F(E) =f(b;f)jb2Bandf= (f₁;:::;f_n) isabasisforp^º1(b)g
be thespaceof framesofE,and letq:F(E)!Bbe givenbyq(b;f) =b.Showthat:
(a)q:F(E)!Bisaprincipal GL(n;R)-bundle.
(b)Thegivenvector bundleisassociatedtoits framebundle viathenatural actionof
GL(n;R) onRⁿ, i.e.,E=F(E)£_GL(n;R)Rⁿ.