By J.J. Duistermaat

Reprinted because it initially seemed within the 1990s, this paintings is as an inexpensive text that can be of curiosity to a number researchers in geometric research and mathematical physics. The book covers a variety of ideas primary to the research and purposes of the spin-c Dirac operator, using the warmth kernels idea of Berline, Getzlet, and Vergne. precise to the precision and readability for which J.J. Duistermaat used to be so popular, the exposition is classy and concise.

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**Extra resources for The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator **

**Example text**

L\1, Z/2Z) is equal to the reduction modulo two of some c E H2 (M, Z). 2. 1,iv)] proved that every four-dimensional compact oriented manifold M satisfies this condition; it therefore can be provided with a spin-c structure. On the other hand, it is a result of Wu Wen-Tsun [76, Th. 74] that a four-dimensional compact oriented manifold M can be provided with an almost complex structure, if and only if there exists a class 4 2 2 C E R (M, Z)suchthatw2 == cmod2andc == 2e+PI. Heree E H (M, Z) and PI E H 4 (M, Z) are the Euler class and the first Pontryagin class of M, respectively.

2. 6) with Vj E V and k :::; q. 6) with k even and with k odd will be denoted by C+ (V) and C- (V), respectively. 8) C+(V) . 9) C+(V) . 11) C-(V) . C-(V) c C+(V). 12) == C-(V) . C+(V) This means that C(V) is a superalgebra; not supercommutative as soon as Q =I O. We also have C::;l(V) n C-(V) V and C::S 1 (V) n C+(V) == R. In particular, C+ (V) is a subalgebra of C(V), and the invertible elements in it form a Lie group, which will be denoted by C+ (V) x. The basic property of the algebra C(V), which actually determines it up to isomorphism, is the following.

12): E == E+ E9 E-, ®C) 'c E+ C E+, (C+(V) ® C) 'c E- c E-, (C-(V) ® C) 'c E+ C E-, (C-(V) ® C) 'c E- c E+. 31) In particular, E+ and E- are C+ (V) 0 C-modules. 40). In the computations, we will meet the supertrace strc A :== tracec A++ - tracec A-lIn [9, p. 110], the choice for c(~) is which has the same square. 32) and i (Q~), Chapter 3. Clifford Modules 28 of endomorphisms A of E. Here we denoted by Ajk the restriction of A to E k , followed by the projection from E to Ej along the complementary subspace.