Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #5
Due 2/9/2000
I. Required problems.
1. Suppose Mis a smooth compact manifold.
(a) If f:Mโโ Ris a smooth function, show that fโvanishes at some point of
M.
(b) Show that there is no smooth submersion F:Mโโ Rkfor any k.
2. Consider the map F:R4โโ R2defined by
F(x, y, s, t)=(x2+y, x2+y2+s2+t2+y).
Show that (0,1) is a regular value of F, and that the level set Fโ1(0,1) is diffeo-
morphic to S2.
3. Exercise 3.4 in the notes.
4. Define subsets of R2by
M={(x, y)โR2:xy =0},
N={(x, y)โR2:x2=y3}.
Answer the following questions for each of these two subsets. Prove your answers
correct.
(a) Is it an embedded submanifold of R2?
(b) If the answer to (a) is no, can it be given a smooth manifold structure (i.e.,
manifold topology and smooth structure) such that it is an immersed sub-
manifold of R2?
II. Optional problems.
5. Let F:Mโโ Nbe a smooth map of constant rank k,andletS=F(M). Show
that Scan be given a manifold topology and smooth structure such that it is an
immersed k-dimensional submanifold of Nand F:Mโโ Sis smooth. Are these
structures unique?
6. Decide whether each of the following statements is true or false, and discuss why.
(a) If F:Mโโ Nis a smooth map and Fโ1(c) is an embedded submanifold of
Mfor some cโN,thencis a regular value of F.
(b) If SโMis a closed embedded submanifold, there is a smooth map F:M
โโ Psuch that Sis a regular level set of F.
