Relations among characteristic classes and fixed points, I: The recognition principle

In this paper we consider relations between characteristic classes and fixed point sets of group actions. The first such example of such a relation is Hopfʹs theorem relating the zeroes of a vector field on a manifold (fixed points of an action of R) to the Euler characteristic of the manifold. More recent examples are given by the theorems of Atiyah and Segal (1968), Baum and Cheeger (1969), Bott (1967), Bott and Baum (1970), Gَmez (1982), Alamo and Gَmez (1989), Daccach and Wasserman (1985,1984), Jeffrey and Kirwan (1995), Guillemin and Kalkman (1996), Quillen (1971) and Witten (1982). Such theorems are called residue theorems or localization theorems because they relate a global invariant of a manifold to local invariants of the fixed point sets. An excellent exposition of this point of view is given by Atiyah and Bott (1984).