PMATH 365: Smooth Manifolds
Assignment 3; due Monday, 11 February 2013
[1] Construct innitely many distinct smooth structures on the topological 1-manifold R.
[2] Consider the function f : R R dened by
1
f (x) =
e x2
0
x > 0,
x 0.
[a] Show that f is smoot
PMATH 365: Smooth Manifolds
Assignment 1; due Friday, 18 January 2013
[1] Let (X1 , T1 ) and (X2 , T2 ) be topological spaces and let f : X1 X2 be a bijection. Show that f is
a homeomorphism if and only if f (T1 ) = T2 , in the sense that U T1 if and only
PMATH 365: Smooth Manifolds
Assignment 2; due Wednesday, 30 January 2013
[1] Let M be a topological n-manifold with boundary. Give the manifold interior M and the manifold
boundary M of M the subspace topologies. You may also use, without proof, the fact
PMATH 365: Smooth Manifolds
Assignment 6; Solutions
[1] We know that both V V and L(V, V ) have dimension n2 . Thus, if we can nd an injective linear
map between these spaces, then it will be an isomorphism. Dene a map Y : L(V, V ) V V
as follows. Let T L
PMATH 365: Smooth Manifolds
Assignment 8; Solutions
[1] Consider M = Rn+1 \ cfw_0. Let = dx1 dxn+1 be the standard orientation form on M . Let
R = xk xk be the radial vector eld. This is a smooth vector eld on M . Let : S n M be the
inclusion, and dene =
PMATH 365: Smooth Manifolds
Assignment 7; Solutions
[1] Let M n be an orientable smooth manifold. Let U be an open subset of M . Then U is a smooth
manifold of the same dimension as M , and the inclusion : U M is an immersion, so the
pushforward at p is a
PMATH 365: Smooth Manifolds
Assignment 3; Solutions
[1] We want to construct innitely many distinct smooth structures on the topological 1-manifold R. A
smooth structure (also called a maximal smooth atlas) is determined by a smooth atlas, which is a
cove
PMATH 365: Smooth Manifolds
Assignment 5; Solutions
[1] Let M be a set, and let cfw_U ; A be a collection of subsets of M . For each A, let : U Rn
be an injective map. Suppose that the following ve conditions are satised.
(i) The set (U ) is open in Rn fo
PMATH 365: Smooth Manifolds
Assignment 1; Solutions
[1] Let f : X1 X2 be a bijection. Let h = f 1 : X2 X1 denote the inverse of f .
If U is a subset of X1 , then f (U ) = h1 (U ) is a subset of X2 . We see that f (U ) is open in X2 for all
U open in X1 if
PMATH 365: Smooth Manifolds
Assignment 2; Solutions
[1] Both M and M are subsets of M , given the subspace topology. Hence they both inherit the
properties of Hausdorness (Hausdorty?) and of existence of a countable basis, since M has these
two properties
PMATH 365: Smooth Manifolds
Mid-term Test; Solutions
[1] Let X and Y be a topological spaces, where X is compact and Y is Hausdor. We proved in class that
any closed subset F of X is compact, and that any compact subset C of Y is closed. Let f : X Y
be a
PMATH 365: Smooth Manifolds
Assignment 4; Solutions
[1] Let f : M N be a smooth map between smooth manifolds. Recall that for p M , we dened the
pushforward (f )p : Tp M Tf (p) N by the rule
(f )p (Xp )(h) = Xp (h f )
for any Xp Tp M and any h C (N ).
[a]