Differential Topology - Computer Graphics and Geometric Modeling - page 531

Proof. See [Hirs76]. Hirsch also explains why one needs the strange constraint

on r.

[Hirs76] gives a number of applications of this theorem and Theorem 8.11.3. For
example, one can use it to give a fairly simple proof of the fact that there is no dif-
ferentiable retraction f : M Æ∂ M (see Exercise 8.11.1 and compare this result with
Theorem 7.5.1.4) and the Brouwer fixed-point theorem that every differentiable map
f: D n Æ D n has a fixed point. The standard proofs of the continuous versions of these
theorems involve algebraic topology. We shall sketch two applications of Theorem
8.11.3 and the Morse-Sard Theorem. The first results in an alternate definition of the
degree of a map and the second deals with intersection numbers, vector fields, and
the Euler characteristic. The transversality approach to these results will give us addi-
tional insights into the underlying geometry.
Let ( M n ,s) and ( N n ,m) be closed compact connected oriented differentiable man-
ifolds and assume that

n n

f : MN Æ

is a differentiable map. By the Morse-Sard theorem, f has regular values. Let q ΠN
be one such regular value for f. By Theorem 8.11.3 and the compactness of M , f -1 ( q )
consists of a finite number of isolated points. Let p Πf -1 ( q ).

Definition. The degree of f at p , deg p f, is defined by

deg p f = +1, if Df( p ) maps T p ( M ) in an orientation preserving way to T q ( N )

= -1, if Df( p ) maps T p ( M ) in an orientation reversing way to T q ( N ).

Define the degree of f over q , deg(f, q ), by

 1

() =

- 1

() π

deg f , q deg f , if f q f ,

p

-

pq Πf

(

)

= 0 , otherwise .

Note that since the set f -1 ( q ) is finite, the sum in the definition of deg(f, q ) is finite.
Figure 8.35 attempts to show what we are doing in the case of a map f : S 1 Æ S 1 . The
map f indicated in the figure has f -1 ( q ) = { p 1 , p 2 , p 3 }, so that, with the orientations as
shown,

( ) =-+=

deg f, q 111 1 .

Observe also that if we assume that f( S 1 ) is a rubber band and pull it tight, then the
two point p 2 and p 3 in the pre-image of q disappear but the degree over q of the new
map is still 1. This “pulling tight” corresponds to changing the map f by a homotopy
and should suggest that homotopies do not change the degree.

8.11.7. Example. Define f : S 1 Æ S 1 by f( z ) = z n , z Œ C . Then deg(f,1) = n
because the n nth roots of unity map to 1 in an orientation-preserving way (Exercise
8.11.2).