Let $f,g:{M}_{1}\to {M}_{2}$ be maps where ${M}_{1}$ and ${M}_{2}$ are connected triangulable oriented n-manifolds so that the set of coincidences ${C}_{f,g}=x\in {M}_{1}|f\left(x\right)=g\left(x\right)$ is compact in ${M}_{1}$. We define a Nielsen equivalence relation on ${C}_{f,g}$ and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if ${M}_{2}={\tilde{M}}_{2}/K$ where ${\tilde{M}}_{2}$ 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 ${M}_{1}$ and ${M}_{2}$ are compact, f...

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...

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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