A fixed point index for equivariant maps

Abstract

The purpose of the paper is to define a fixed point index for equivariant maps of G-ENR's and to state and prove some of its properties, such as the compactly fixed G-homotopy property, the Lefschetz property, its converse, and the retraction property. At the end, some examples are given of equivariant self-maps which have a nonzero index (hence cannot be deformed equivariantly to be fixed point free) but have a zero G-Nielsen invariant.

A Note on Equivariant Fixed Point Theory

“…This idea, which dates back to K. Wilczynski [47], K. Komiya [27,28] and W. Marzantowicz [35], has been also recently used by Zhao [55] in the definition of a fixed point index for maps of pairs and by the author in [15] for the definition of the index that we are describing in this section. The steps for the definition are the following.…”

“…The equivariant index I G (f ) is the collection of Reidemeister traces parameterized as follows. For every H ∈ Iso(X) let I G (f )(H) be the Reidemeister trace L(f H ) ∈ ZR(f H ) of the restriction f H : X H → Y H of the G-map f : X → Y to the subspace X H ⊂ X H (see remark (6.4) below for a comment on the Reidemeister trace / generalized Lefschetz number for local non-connected maps; see also (6.3) for the differences with the results of [15]). Since a G-homotopy which is G-taut for every t ∈ I preserves the traces L(f H ), it is proved that I G (f ) can be defined for all compactly fixed G-maps, as the index of any G-taut -approximation (see (4.3)).…”

“…Here we recall briefly the core of example 5.2 of [15]. Let G be the group of order 2 acting on R 3 by (x, y, z) → (−x, y, z).…”

“…Thus N (f G ) = N (f ) = 0. On the other hand, even considering the regular neighborhood of X 0 in R 3 (or in any bigger euclidean space), it is not possible to deform f to a G-map which is fixed point free, since the equivariant fixed point index I G (f ) of [15] is not zero. For a similar example see also Vidal and Izydorek in [25].…”

“…A combinatorial way to encode the information about the number of classes of a given index is to associate to a map (or to its Reidemeister trace) the function which assigns to a fixed point class its fixed point index. As remarked in 3.1 of [15], the set of isomorphism classes of functions with values in Z 0 and finite domain is a commutative monoid, with respect to the disjoint union and the cartesian product, so that the Grothendieck ring R is well-defined. This concept, mutated from the Burnside ring definition and applied to the context of Nielsen fixed points, allows to define an index which is in an intermediate position between the Nielsen number (which counts the essential fixed point classes) and the Reidemeister trace (which counts the essential fixed point classes, together with their fixed point indices, together with their coordinates in the Reidemeister twisted conjugacy classes).…”