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.

We show that the property of having only vanishing triple Massey products in equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular it can be viewed as a partial answer to a question posed by Allday and Puppe about finding conditions ensuring the "formality" of G-actions.

Let n be an integer with n ≥ 2 and $X_i$ be an infinite collection of (n-1)-connected continua. We compare the homotopy groups of $\Sigma \left({\prod }_i{X}_i\right)$ with those of ${\prod }_i\Sigma X_i$ (Σ denotes the unreduced suspension) via the Freudenthal Suspension Theorem. An application to homology groups of the countable product of the n(≥ 2)-sphere is given.

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_2$–cohomology groups $H^j\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

The purpose of this note is to prove the converse of the Lefschetz fixed point theorem (CLT) together with an equivariant version of the converse of the Lefschetz deformation theorem (CDT) in the category of finite G-simplicial complexes, where G is a finite group.

Let p be a prime number and X a simply connected Hausdorff space equipped with a free ${\mathbb{Z}}_p$-action generated by $f_p:X\to X$. Let $\alpha :S^{2n-1}\to S^{2n-1}$ be a homeomorphism generating a free ${\mathbb{Z}}_p$-action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map $F:(S^{2n-1},\alpha )\to (X,f_p)$. As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

If $X$ is a $n$-dim polyhedran, then using geometric techniques, we construct groups $H_p(X{)}_{\Delta }$ and $H^p(X{)}_{\Delta }$ such that there are natural isomorphisms $H^p(X{)}_{\Delta }\simeq H_{n-p}\left(X\right)$ and $H_p(X{)}_{\Delta }\simeq H^{n-p}\left(X\right)$ which induce an intersection pairing. These groups give a geometric interpretation of two spectral sequences studied by Zeeman and allow us to prove a conjecture of Zeeman about them.