CHAPTER 1
Index Theor y
The subjec t o f thi s lectur e serie s interrelate s a numbe r o f difFeren t area s o f
mathematics, bu t i n m y min d i t bega n wit h inde x theory , s o tha t i s wher e I
will start .
Let M b e a compac t (smooth ) manifold . The n we have the de Rham comple x
n°(M) -i n\M) 4 .. . 4 fi n(M),
in whic h d i s th e exterio r derivative , an d becaus e d
2
= 0 w e ca n for m it s
cohomology, th e de Rham cohomology o f M. Notic e tha t th e vecto r space s
Vtx{M) ar e infinite-dimensional , bu t w e kno w tha t th e d e Rha m cohomolog y
is isomorphi c t o th e ordinar y cohomology , an d i s therefor e nnite-dimensionai .
When formulate d i n term s o f d e Rha m cohomolog y thi s i s a resul t abou t th e
Solution space s o f certain partia l differentia l equations , an d w e might loo k fo r a
proof o f it i n term s o f analysis.
Analysis o f Dira c Operator s
Let M b e an y manifold (compac t o r not), and le t fi* (M) denot e the spac e of
compactly supported i-form s o n M . Th e first thin g we need t o d o is to complet e
the spaces ft l
c
(M) o f smooth differentia l form s t o Huber t spaces , which ar e muc h
more tractabl e fro m th e perspectiv e o f functiona l analysis . Le t u s choos e a
Riemannian metri c o n M. Thi s metri c define s a positiv e measur e p n M , an d
it als o gives ris e t o a Hermitia n inne r produc t o n th e cotangen t bündl e o f M ,
and henc e o n al l it s associate d exterio r powers . W e ma y therefor e dehn e a n
inner product (• , •), called th e L
2
inne r product, o n each of the spaces ü
l
c(M) b y
integrating th e loca l inner product s (• , •) with respec t t o the measur e JJL:
*,/?= / (a(x) )ß(x))dß(x).
JM
By completing the space s il l
c
(M) i n this inner product , w e obtain Huber t space s
Q%L2{M)
o f Square integrable forms.
The d e Rha m comple x no w become s a comple x o f Huber t space s an d un -
bounded Operators . W e recall tha t a n unbounded Operator T o n a Hubert spac e
l
http://dx.doi.org/10.1090/cbms/090/01
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