Generalized Jiang and Gottlieb groups

Given a map {f\colon X\to Y} , we extend Gottlieb’s result to the generalized Gottlieb group {G^{f}(Y,f(x_{0}))} and show that the canonical isomorphism {\pi_{1}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}(Y)} restricts to an isomorphism {G^{f}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}^{\tilde{f}_{0}}(Y)} , where {\mathcal{D}^{\tilde{f}_{0}}(Y)} is some subset of the group {\mathcal{D}(Y)} of deck transformations of Y for a fixed lifting {\tilde{f}_{0}} of f with respect to universal coverings of X and Y, respectively.

A Homotopical Conner-Raymond Theorem and a Question of Gottlieb

A homotopy theoretic version is given of the following result of Conner and Raymond: If the circle acts on a space so that the orbit map induces an injection in homology, then the space fibres over the circle with finite structure group. This homotopical analogue is related to recent results pertaining to the effect of the fundamental group's structure on the Euler characteristic. It is also used in the construction of a compact, simple 7-manifold with trivial Gottlieb group which, together with an infinite dimensional example of Ganea, answers a question of Gottlieb.