The Lefschetz Principle , Fixed Point Theory , and Index Theory

This is a rough historical account of some uses of the Lefschetz Principle in fixed point theory and index theory. The Lefschetz Principle states that the alternating sum of the traces on cohomology (a global and rigid invariant) is equal to the alternating sum of the traces on the underlying cochain complex (a local and far less rigid invariant). The original Lefschetz Theorem for compact polyhedra then follows easily. The Lefschetz Principle extends readily to index theory and general fixed point theory on compact manifolds, where it is more commonly known as the heat equation method. We outline the proofs of the Atiyah-Singer Index Theorem and the Atiyah-Bott Fixed Point Theorem using this method. Some Bott Magic? “No! Just physics!” The above photo was taken at the Bott house on Martha’s Vinyard, probably in the summer of 1983, and probably by Paul Schweitzer, SJ. Pictured with the Master are the author in the middle, and Lawrence Conlon (a Bott student). Note the charcoal starter, home made from a stove pipe and complete with floppy wooden handles. Someone asked Raoul if it worked by magic. His answer was typical of the man–succinct and to the point. This is an expanded write up of the talk I gave at the conference “A Celebration of Raoul Bott’s Legacy in Mathematics”, held June 9-13, 2008, at the Centre de Recherches Mathématiques, Université de Montréal. Its purpose was to illustrate some of the magic that Raoul Bott worked in mathematics. It covers some very deep results, so time constraints dictated that proofs be outlined only in the broadest terms. They should not be taken literally. Readers interested in the details of the proofs should consult the references. 1 2 J. L. HEITSCH NOVEMBER 5, 2009 1. Lefschetz Fixed Point Theorem The classical Lefschetz Fixed Point Theorem gives a cohomological criterion for a continuous map to have a fixed point. In its simplest form its proof is almost obvious once the Lefschetz Principle is proven. The material in this section follows closely a lecture given by Raoul Bott in 1984. Theorem 1.1 (Lefschetz Fixed Point Theorem). Let X be a compact polyhedron, and f : X → X a continuous map. Set L(f) = ∑ k (−1) tr ( f∗ : H(X)→ H(X) ) . If L(f) 6= 0, then f has a fixed point. Since the map f∗ only depends on the homotopy class of f , L(f) is a homotopy invariant. Also note that L(f) is a generalization of the Euler number χ(X) of X, since L(I) = χ(X), where I : M → M is the identity map. A good example to keep in mind is the antipodal map A : S → S, which has no fixed points, so L(A) must be zero. In particular, tr ( A∗ : H(S) → H(S) ) = 1, since S is connected, and because A is orientation reversing, tr ( A∗ : H(S)→ H(S) ) = −1, so indeed L(A) = 0. Proof. Our method of proof is to assume that f has no fixed points and then show that L(f) = 0. The most naive approach would be to attempt to show that all the individual tr ( f∗ : H(X) → H(X) ) are zero. As the example above shows, this is a vain hope, but it does contain the essential idea of the proof, once we realize that the reason it doesn’t work is that the spaces H(X) are too small. If we are willing to expand the domain of the f∗ the proof becomes obvious. To do this we need Proposition 1.2 (The Lefschetz Principle). L(f) = ∑ k (−1) tr ( f∗ : C(X)→ C(X) ) . Here we are using simplicial cochains C(X) to compute the cohomology of X, and we may use simplices as small as we like to do so. Since f has no fixed points and X is compact, f must move points at least a fixed positive distance, say δ. Then we use simplices which have diameter less than δ/100 and we approximate f by a simplicial map g so that pointwise g is within δ/10 of f . We may assume that δ is so small that g is homotopic to f , so we may as well assume that g = f . Because f moves points at least δ and our simplices are at most δ/100 in diameter, it is impossible for f to map any simplex to itself, so it is immediate that for all k, tr ( f∗ : C(X)→ C(X) ) = 0. Proof of The Lefschetz Principle. Denote by Z the cocycles in C = C(X), and by B the coboundaries. Then we have the following commutative diagrams of finite dimensional vector spaces. (∗) 0 −→ Z −→ C −→ B −→ 0 0 −→ Z −→ C −→ B −→ 0 ? ? ? (∗∗) 0 −→ B −→ Z −→ H −→ 0 0 −→ B −→ Z −→ H −→ 0 ? ? ? where all the vertical maps are restriction of f∗. Note that the rows of both diagrams are short exact sequences. Thus for all k, tr ( f∗ |Z ) + tr ( f∗ |B ) = tr ( f∗ |C ) and tr ( f∗ |B ) + tr ( f∗ |H ) = tr ( f∗ |Z ) . THE LEFSCHETZ PRINCIPLE, FIXED POINT THEORY, AND INDEX THEORY 3 We may combine all this information as follows. For t ∈ R, set Bt = ∑ k t tr ( f∗ |B ) , Ct = ∑ k t tr ( f∗ |C ) , Ht = ∑