Abstract
The rank of the image of the Gottlieb group under the Hurewicz map is called the h-rank of a space. We give an upper bound on the h-rank of principal bundles over tori; in particular, we use this bound to show that certain families of these bundles consist of nonsimply connected simple manifolds with trivial Gottlieb group (thus answering a question of Gottlieb in the negative). We then show that this implies the existence, for each of the examples, of self-homeomorphisms which induce the identity on homotopy groups, but which are not based isotopic to the identity map.
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