Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups (Q1295246)

In [the reviewer, ``Some research problems on homotopy-self-equivalences'', Lect. Notes Math. 1425, 204--207 (1990; Zbl 0704.55005), p. 206] the basic problem of finding the relation between the homotopy-self-equivalences which fix homology and those which fix homotopy is presented. The present paper explores questions for special cases of Moore and co-Moore spaces, and Poincaré duality complexes. In the first two cases, the results are complete, as well as including those self-equivalences which fix cohomology. When \(X\) is a Poincaré duality complex, the self-equivalences which fix homology are those which fix cohomology. The authors close with some very interesting examples. If \(X=\mathbb{C} P^n\vee S^{2n}\), the self-equivalences which fix homotopy through dimension \(2n\) is \(\mathbb{Z}\), while those which fix homotopy through dimension \(2n+1\) is the trivial group.