Equivariant fixed-point indices of iterated maps

Abstract

We describe an equivariant version (for actions of a finite group G) of Dold's index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered. Mathematics Subject Classification (2000). Primary 37C25, 55M20, 55N91; Secondary 55P91.

The homotopy coincidence index

“…The null-set Null(s) is the fixed-subspace Fix(f ) = {x ∈ B | f (x) = x} of f and h-Null(s) is naturally identified with the homotopy fixed-point set h-Fix(f ) (see [3,5]) consisting of the paths α : [0, 1] → N such that α(1) = f (α(0)).…”

“…In [5] and [3] we defined, in a general fibrewise and equivariant setting, the homotopy Lefschetz index…”

“…Indeed, we may take V = 0 and j = 1 : B → N . Then the construction reduces to that described in [5,3]. The proof is completed by comparing the two definitions, taking i in Definition 5.2 to be a smooth embedding, W to be a tubular neighbourhood and W to be U × V.…”