4 1. INDE X THEOR Y

the Operator o f convolution b y the invers e Fourier transfor m o f /. I n particular ,

suppose /(A ) = e~ A2/2. The n

f(D)u{x) = -j=J_ e-^-y?'

2u{v)dy,

in other words, f{D) i s a smoothing Operator with Gaussian kernel. A smoothing

Operator wit h compactl y supporte d kerne l i s a compac t Operator , s o wheneve r

f{D) i s cu t dow n t o a compac t domai n b y multiplicatio n b y a compactl y

supportedp , we get a compact Operator . Tha t is , f(D) i s locally compact .

Let's confin e ou r attentio n fo r th e moment , however , t o th e cas e that M i s a

compact manifold. The n ellipti c regularity has the following simpl e consequence :

the comple x fi£

2

i s the direc t su m o f tw o close d subcomplexe s invarian t unde r

Z), th e comple x W * = kerD o f Harmonie form s (o n whic h th e differentia l d i s

identically zero) , an d it s orthogona l complemen t H

±]

moreover , th e restrictio n

of D t o

H1-

i s invertible, an d this implies that H

1-

i s an aeyclic complex — it ha s

trivial cohomology . Thu s w e ge t th e Hodge theorem H

l

DR(M) = W(M). Sinc e

W(M), a s an eigenspace of a generalized Dirac Operator on a compact manifold ,

must b e finit e dimensional , thi s give s u s the analyti c explanatio n tha t w e were

seeking for th e finite-dimensionality o f de Rham cohomology .

Index theor y

The notion of index is abstractly defined i n functional analysi s for all Fredholm

Operators: a Predholm Operato r fro m on e Hubert spac e to anothe r i s a bounde d

Operator which is invertible modul o th e compac t Operators . Suc h a n Operato r T

has kerT an d kerT * finite-dimensional, an d it s index is, by definition, Ind(T ) =

dim ker T — dim ker T*. T o relate thi s definitio n t o th e Dira c Operator s tha t w e

have bee n considering , le t u s introduc e th e notio n o f a chopping funetion.

DEFINITION 1.5: A choppin g funetio n is a continuous odd funetion x : R — •

[—1,1] such that x(t) — • ± 1 as t —* ±oo .

LEMMA 1.6: Let D be a generalized Dirac Operator on a compact manifold M.

Then x{D) i$

a

Fredholm Operator, and if xi and X2 dre two different chopping

functions, then the corresponding Fredholm Operators xi {D) and X2 (D) differ by

a compact Operator.

PROOF: Sinc e x

1

— 1 an d xi ~ X2 ar e function s vanishin g a t infinity , th e

corresponding Operator s ar e compac t b y 1.3. D

Since

x{D)ls

self-adjoint , th e index of

x{D)1S

zero . However , Dirac Operator s

on even-dimensiona l manifold s frequentl y com e equippe d wit h a piec e o f extr a

strueture called a grading: thi s is a self-adjoint involution

3

e which anticommute s

An involution i s a n Operato r whos e Squar e i s 1.