Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #2
Due 1/19/2000
I. Required problems.
1. Compute the coordinate representation for each of the following maps, using stere-
ographic coordinates for spheres; use this to conclude that each map is smooth.
(a) ฮน:SnโRn+1 is inclusion.
(b) A:SnโSnis the antipodal map A(x)=โx.
(c) F:S3โS2is given by F(z,w)=(zw +wz, iwzโizw, zzโww), where we
think of S3as the subset {(w, z):|w|2+|z|2=1}of C2.
2. Let M=B1(0), the closed unit ball in Rn. Show that Mis a manifold with
boundary, and can be given a smooth structure in such a way that the inclusion
map M,โRnis smooth.
3. Let Mbe a smooth n-manifold with b oundary. Show that Int Mis a smooth
n-manifold and โM is a smooth (nโ1)-manifold (both without boundary).
II. Optional problems.
4. Let A1and A2be the atlases for Rdefined by A1={(R,Id)},andA2={(R,ฯ)},
where ฯ(x)=x3.Letf:RโRbe any function. Determine necessary and
sufficient conditions on fso that it will be:
(a) a smooth map (R,A2)โ(R,A1);
(b) a smooth map (R,A1)โ(R,A2).
5. For any topological space X,letC(X) denote the vector space of continuous
functions f:XโR.IfMand Nare topological manifolds and F:MโNis a
continuous map, define Fโ:C(N)โC(M)byFโ(f)=fโฆF.
(a) Show that Fโis linear.
(b) Show that Fis smooth if and only if Fโ(Cโ(N)) โCโ(M).
(c) If Fis a homeomorphism, show that Fis a diffeomorphism if and only if
Fโ:Cโ(N)โCโ(M) is an isomorphism.
Thus in a certain sense the entire smooth structure of Mis encoded in the space
Cโ(M).
