Coincidence theory for spaces which fiber over a nilmanifold.
On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem
Let be maps where and are connected triangulable oriented n-manifolds so that the set of coincidences is compact in . We define a Nielsen equivalence relation on and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if where is a connected simply connected topological group and K is a discrete subgroup then all the Nielsen coincidence classes of f and g have the same coincidence index. In particular, when and are compact, f...
Corrections to "On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem" (Fund. Math. 140 (1992), 191–196)
Equivariant Nielsen theory
Fixed point theory for homogeneous spaces, II
Let G be a compact connected Lie group, K a closed subgroup and M = G/K the homogeneous space of right cosets. Suppose that M is orientable. We show that for any selfmap f: M → M, L(f) = 0 ⇒ N(f) = 0 and L(f) ≠ 0 ⇒ N(f) = R(f) where L(f), N(f), and R(f) denote the Lefschetz, Nielsen, and Reidemeister numbers of f, respectively. In particular, this implies that the Lefschetz number is a complete invariant, i.e., L(f) = 0 iff f is deformable to be fixed point free. This was previously known under...
Equivariant evaluation subgroups and Rhodes groups
A note on generalized equivariant homotopy groups
In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce Γ-Rhodes groups, where Γ admits a certain co-grouplike structure. Evaluation subgroups of Γ-Rhodes groups are discussed.
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