Topology Qualifying Exam, August 2012
Instructions: Do all problems. Policy on misprints: If you feel that a problem has
been misstated, then restate it in the way you believe it should be stated and solve
the restated problem. Do not restate the problem so as to make the problem trivial.
1. (a.) Give a careful definition of the tangent bundle of a manifold X.
(b.) Show that the total space of the tangent bundle of S2is not diffeomorphic to
S2ĆR2.
2. Suppose Mis a smooth nāmanifold, and f:MāR2n+1 is injective. Let Bbe a
closed subset of M, let Ube an open subset containing B, and ĻBa smooth bump
function for Bsupported in U. For any bāR2n+1, define gb:MāR2n+1 by
gb(x) = f(x) + b ĻB(x).
Show that there is some b6= 0 in R2n+1 so that gbis injective.
3. Let Īøand Ī·be smooth 3āforms on the 7ādimensional sphere S7. Prove:
ZS7
dĪø ā§Ī·=ZS7
Īøā§dĪ·
4. Let f:S1āS1be given by f(z) = z2, where we regard S1as the unit circle in C.
Find a presentation of the fundamental group of the mapping torus of f:
Mf=S1ĆI/(z , 1) ā¼(f(z),0)
5. A knot Kis a (smoothly) embedded copy of S1in S3. For a given knot K, the knot
complement is the space S3rK. Use the MayerāVietoris sequence to compute the
reduced homology groups of any knot complement. [Hint: One way to do this is to
write Kas the union of two embedded copies of the interval (0,1). Take Ato be
S3minus one of these, Bto be S3minus the other.]
6. Let Mbe a closed connected 5āmanifold such that Ļ1(M)ā¼
=Z/7Z. If H2(M;Z)ā¼
=
Z, compute all other homology and cohomology groups of Mwith integral coeffi-
cients. State all theorems that you use.
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