Fixed Points and Coincidences in Torus Bundles

Abstract

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper, we investigate fiberwise analoga and present a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace E B and a natural generalization of Nielsen numbers.As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They also allow a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S 1 ) 2 -bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.

Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

“…In the case when the fibres F M and F N are tori of arbitrary, possibly different, dimensions greater than or equal to 1, and the gluing maps A M and A N are Lie group automorphisms, Questions 1.1-1.3 and the whole related Nielsen coincidence theory have been recently reduced to simple, purely algebraic problems (which, nevertheless, can be highly non-trivial) (see [11,12]). In the proofs theω B -invariant turned out to be absolutely crucial.…”

“…Since N is also a torus bundle we can use the techniques of [12] together with Toda's tables of stable homotopy groups of spheres (see [16]). The Lie group structure on the fibres makes F into an abelian group.…”

“…Here, N is a torus bundle as in [12], with a well-defined group structure (via complex multiplication) on each one-dimensional fibre. Indeed, the gluing map A N (see (1.3)) is the identity map id or complex conjugation, respectively.…”

“…In this paper we follow [13,14] and define N B (f 1 , f 2 ) to be the number of non-trivial summands in this sum decomposition (and not, as in [3], in the analogous decomposition forω B (f 1 , f 2 )). [3] and [12]. If m > n = 2, we can still add and subtract f 1 , f 2 in a fibrewise fashion.…”

“…, k, which are pairwise Nielsen inequivalent (see [3,Definition 4.1] and [12]). But the coincidence manifold C(s k • p M , a • f 0 ) is empty (if k = 0) or else consists of the (connected) fibres {(2j − 1)/2k} × S m−1 , j = 1, .…”