Download Topology I – Problem Set Five Fall 2011

Topology I – Problem Set Five Fall 2011
1. Let G be a topological group, and H a discrete subgroup.
(a) Show that G → G/H is a regular covering space.
(b) Assume G is simply connected and prove that π1 (G/H, 1) ∼
= H.
2. Let Gi be a collection of more than one non-trivial group. Prove that their free product
is non-abelian, contains elements of infinite order, and that its center is trivial.
3. Let G , H, G0 , and H 0 be cyclic groups of orders m, n, m0 , and n0 respectively. If G ∗ H
is isomorphic to G0 ∗ H 0 then m = m0 and n = n0 or else m = n0 and n = m0 .
4. Let M and N be n-dimensional manifolds, where n > 2. Let M #N be their connected
sum. Show that π1 (M #N ) = π1 (M ) ∗ π1 (N ).
5. (a) Let F be a free group on finitely many generators. Show that there is a four
dimensional manifold whose fundamental group is F (Hint: the fundamental group
of S 1 × S 3 is free on one generator. Use the connected sum operation and Problem
4 from Problem Set Three.)
(b) Let G be any finitely presented group. Show that there is a 4-manifold that has G as
its fundamental group (Hint: Let F be the free group on the generators, R the set of
relations. Start with X, the 4-manifold from part a). Represent each relation in R
by a loop in X. Go ahead and assume that the loop can is a smooth, simple closed
curve. Replace the loop by a tubular neighborhood N , which is homeomorphic to
S 1 × D3 . The boundary of N is homeomorphic to S 1 × S 2 . Delete the interior of
N . Now S 1 × S 2 is also the boundary of D2 × S 2 , so take a copy of D2 × S 2 and use
it to fill in the ’hole’ by identifying on the boundary. Now use Seifert Van-Kampen
to show that you have killed off the generator.)
6. Let p : R2 → T be the covering map p(x, y) = (e2πix , e2πiy ) and let L be the subspace
(grid) of R2 consisting of all those points with x-coordinate an integer, or y-coordinate
an integer. Let X be p(L) ⊂ T . Then X is the ’figure-eight space’ and the fundamental
group of X is a free group on two generators.
(a) Show that p|L : L → X is a covering space.
(b) Choose basepoints (0, 0) and (1, 1) for L and X. What is π1 (L, (0, 0)) ?
(c) Show that (p|L )∗ (π1 (L, (0, 0))) is the commutator subgroup of π1 (X, (1, 1)).
7. Let q : X → Z be a covering space. Let p : X → Y be a covering space. Suppose there
is a map r : Y → Z such that q = r ◦ p. Show that r : Y → Z is a covering space.
8. Prove that the fundamental group of a compact, connected surface with boundary is a
free group. Express the rank of this group in terms of the genus and the number of
boundary components.