Fixed-point theory for homogeneous spaces

Abstract

Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p * : H n ( G ) → H n ( M ) is nonzero, where n = dim M . In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M . Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f , which can be computed algebraically.

Fixed point theory for homogeneous spaces, II

“…A simple formula relating L(f, f ) and D K (ϕ, ϕ) is given; it generalizes a similar formula obtained in [16] when K is a finite subgroup. The connection between D K (ϕ, ϕ) and the equivariant obstruction o K (ϕ), similar to that between L(f, f ) and o(f ) as in [4], will also be given.…”

“…The appropriate equivariant local coefficient system admits an action from the equivariant fundamental groupoid whose typical object group is the fundamental group of a transformation group of F. Rhodes [14]. This action coincides with the action of the extension group Γ G on π as described in [16]. We show that o K (ϕ) = 0 iff o(f ) = 0.…”

“…We call such spaces Jiang type spaces. In [16], we employed the Nielsen root theory of R. Brooks [1] to further explore the connection, made explicit by E. Fadell in [3], between the fixed point problem for orientable homogeneous spaces and the associated Borsuk-Ulam type problem. In particular, we showed that an orientable homogeneous space M = G/K, where G is a compact connected Lie group and K a closed subgroup, is a Jiang type space provided the homomorphism p * : H n (G) → H n (M ) induced by the projection map on the n = dim M -th integral homology is nonzero.…”

“…Thus p cannot be homotopic to any map q with q −1 ([e]) = ∅. For the map ϕ, the ordinary root index ω(ϕ, S 3 ) (see [1] or [16]), which can be viewed as the primary obstruction to deforming ϕ to be root free, is zero. Furthermore, the secondary obstruction, which lies in H 3 (S 3 ; π 3 (S 2 )), is also trivial.…”

“…This simple example suggests that the hypothesis p * = 0 need not be necessary and equivariant obstruction may be used to handle the situation when p * is zero. Since the assumption p * = 0 was used in [16] to show that the ordinary root index was nonzero, which then implied the equivariant root index to be nonzero, we must by-pass the use of the ordinary root index and instead give a direct proof of the nonvanishing of the equivariant root index.…”