CHAPTER 1

Introduction

Here are two sample theorems that we are going to prove in these

lectures. To state the first one, we introduce some standard notation.

Given manifolds M and TV, we write

(7r(M,

N) for the set of r times

continuously differentiable maps from M to N (resp. continuous maps if

r = 0). This presupposes, of course, that M and N are at least of class

Cr. We use 'smooth' synonymously with C°°. The tangent bundle

of a differentiable manifold M is denoted by TM. A Cr-immersion

f: M -• T V is an / G

Cr(M,

TV), r 1, whose differential Tf: TM -•

T7V is fibrewise injective. Recall that a manifold M is called closed if

each component of M is compact and without boundary; it is called

open if each component is noncompact or with nonempty boundary.

Usually we shall implicitly assume that our manifolds are connected; in

that case 'open5 and 'closed' are complementary notions. The following

theorem is commonly known as the Smale-Hirsch theorem. It was

proved in the special case M = Sm, T V = IRg, g m + l, byS. Smale,

and in its general form by M. Hirsch. There are other proofs due to

R. Palais and V. Poenaru.

THEOREM

1.1 (Smale-Hirsch). Let M and N be smooth manifolds.

Assume that dimM dim iV or that M is open. Then f G C°(M, N)

is homotopic to a smooth immersion of M into N if and only if f is

covered by a fibrewise injective bundle map F G C®(TM,TN).

To say that / is covered by F means that we have a commutative

diagram

TM —F-^ TN

M —^- TV,

where the vertical maps are the obvious projections.

l