Invariance Theory, the Heat Equation and the Atiyah-Singer by Peter B. Gilkey

By Peter B. Gilkey

This ebook treats the Atiyah-Singer index theorem utilizing the warmth equation, which supplies an area formulation for the index of any elliptic advanced. warmth equation tools also are used to debate Lefschetz mounted element formulation, the Gauss-Bonnet theorem for a manifold with soft boundary, and the geometrical theorem for a manifold with soft boundary. the writer makes use of invariance idea to spot the integrand of the index theorem for classical elliptic complexes with the invariants of the warmth equation.

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1) Since Gl (H) is open, Spec (T) is a closed subset of C. 2) g(A) = L:n> oA - n - l Tn converges in End(H) . Since (T - A)g(A) = g(A)(T - A) = -I, we may conclude that A � Spec (T). Thus Spec (T) � {A : I A I � I IT I I } . 3) Let A � R. 4) I I (T - A)x l l · l l x l l � : (T - A)x · x i � Im (A) I I x l l 2 . L = {0}, T - A is onto. Since S)1 (T - A) = {0}, A � Spec (T). 5) spec(T) C [- I I T I I , I I T I IJ C R. If A E [- I I T I I , I I T I I ] , let E(A) := SJt (T - A) = {x E H : Tx = Ax}.

Remark: Iso (E, F) C Fred (E, F) and Index (T) = 0 if T E Iso (E, F). 4: Let T E Fred (E, F) and S E Fred (F, G). Lemma (a) Index (T*) = -Index (T). (b) Index ( ST) = Index ( S) + Index (T) . (c) Fred (E, F) (d) Index is an open subset of Hom (E, F). : Fred ( E , F) Z is continuous and locally constant: --4 Proof: (a) is immediate from the definition. L }. 4. L ) = Index (T) + Index (S) . 4 Index of Fredholm Operators We prove (c) and (d) as follows. L and F = S)1 (T*) EB 9t (T). L to 9t (T) . 22) S (fo EB e) := 1l'SJ1 {T (e) EB (fo + S(e)).

It is immediate that p has compact U support. 40) § 1 . 8 to see: lq(x, � + (, �) I :SCk (1 + I� + (l ) d (1 + IW - k :SCk (1 + l(l ) d (1 + I W idl - k . 41) :S C(1 + l(i) d . 42) which arise from the given estimates for r show p E Sd so R E w d (U) . 43) The remainder qk decays to arbitrarily high order in (�, ( ) and after in­ tegration gives rise to a symbol of arbitrarily high negative order which may therefore be ignored. 2 to complete the proof of (b) by checking: p(x, ( ) = � 1 <> 1 9 f eix · ed'(q(x, ( , �)� <> d�/a!

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