Math 548 Geometric Structur es Winter 2008
Assignment #3 (CORRECTED)
Due March 19, 2008
For full credit, do any seven of the fol lowing problems.
1. Suppose Mis a smooth manifold and E→Mis a smooth (real or complex) vector bundle. Prove that
there is a vector bundle E0→Msuch that E⊕E0is trivial.
2. Determine classifying spaces for the cyclic groups Zand Z/hni.
3. Suppose G→V→Band G0→V0→B0are principal bundles.
(a) Show that the Cartesian product bundle G×G0→V×V0→B×B0is a principal G×G0-bundle.
(b) If both Vand V0are universal, show that the Cartesian product bundle V×V0is also universal.
(c) Using the fact that every finitely generated abelian group is a direct sum of cyclic groups, deter-
mine a classifying space for each finitely generated abelian group.
4. Recall that a fiber bundle V→Bis said to be n-universal if every bundle with the same group and
fiber over a CW complex (or manifold) of dimension at most nis isomorphic to a pullback of V. I
showed in class that RP2is 1-universal for real line bundles, and CP1is 2-universal for complex line
bundles. Given a bundle E→Mover a manifold or CW complex of dimension at most n, we extend
the notion of classifying map for Eto include a map from Minto the base of an n-universal bundle
Vthat pulls Vback to E.
(a) Show that the map f:S1→RP2given by
f(eiθ) = cos θ
2,sin θ
2,0
is a classifying map for the M¨obius bundle. (Be sure to verify that it is well-defined and continu-
ous.)
(b) Let U→CP1denote the tautological complex line bundle over CP1. For k > 0, show that the
map pk:CP1→CP1given by
pk[z, w] = [zk, w k]
is a classifying map for Uk=U⊗...⊗U; and that
p−k[z, w] = [ ¯zk,¯wk]
is a classifying map for Uk.
5. Given a topological space Xand a basepoint x0∈X, define π0(X, x0) to be the set of path components
of X, considered as a pointed set with the path component containing x0as its distinguished point.
(In general, this set does not have a group structure.)
(a) Given a fiber bundle F→E→B, show that there is a map ∂:π1(B , b0)→π0(F, f0) such that
the homotopy sequence of the bundle extends to an exact sequence
...→π1(E , e0)→π1(B, b0)→π0(F, f0)→π0(E, e0)→π0(B, b0)→ {0},
where exactness is interpreted to mean that the image of each map is equal to the preimage of
the basepoint under the next map.
(b) If π:E→Bis a fiber bundle with path connected fiber, show that the induced homomorphism
π∗:π1(E, e0)→π1(B, b0) is surjective.
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