Fixed Point Theory for Homogeneous Spaces A Brief Survey

Abstract

In classical Nielsen fixed point theory, the computation of the Nielsen number is very difficult in general and is one of the central issues in the field. W. Franz in [Fr] showed that the fixed point classes of any selfmap f of a lens space have the same fixed point index, in which case, the Nielsen number N(f) is a divisor of the Lefschetz number L(f) and is either zero or equal to the Reidemeister number R(f). In [Ji1], B. Jiang gave conditions on the fundamental group under which all fixed point classes have the same index. IfX is a Jiang space, i.e. π1(X) satisfies the so-called Jiang condition, then for all f :X → X, (i) if L(f) = 0 then N(f) = 0 and (ii) if L(f) = 0 then N(f) = R(f). While the class of Jiang spaces include simply connected spaces, Lie groups, H-spaces, generalized lens spaces, and coset spaces of the form G/G0 where G0 denotes a connected closed subgroup of a compact connected Lie group G, such a space necessarily has abelian fundamental group. If π1(X) is finite, it was shown in [Ji1] that (i) and (ii) hold if π1(X) acts trivially on the rational homology of the universal cover of X. A slight generalization of the Jiang condition was also introduced in [FH1] by E. Fadell and S. Husseini. Another computational technique is that for fiber-preserving maps, relating the Nielsen number of a fiber map with the Nielsen numbers of the induced map on the base and of the restriction map on the fiber (see e.g. Chapter 4 of [Ji2] or [HKW]). In 1984, D. Anosov in [An] showed that for every selfmap f :N → N of a compact nilmanifold N , N(f) = |L(f)|. This result extends the same result for selfmaps of tori by Brooks, Brown, Pak and Taylor (see [BBPT].) By a nilmanifold, we mean a coset space N = G/Γ of a nilpotent Lie group G by a closed subgroup Γ. In [NO], B. Norton-Odenthal strengthened Anosov’s result by showing that N(f) > 0 ⇒ N(f) = R(f), employing techniques of [FH2]. Following the work of A. Mal’cev, we may assume that G is simply-connected and thus