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

Abstract

For a based, 1-connected, finite CW-complex X, we study the following subgroups of the group of homotopy classes of self homotopy equivalences of X: E * (X), the subgroup of homotopy classes which induce the identity on homology groups, E * (X), the subgroup of homotopy classes which induce the identity on cohomology groups and E dim+r # (X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ≤ dim X + r. We investigate these groups when X is a Moore space and when X is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on X such that E * (X) = E * (X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine E dim+r # (X) for certain spaces X and completely determine the group when X = S m × S n and X = CP n ∨ S 2n. 1991 Mathematics Subject Classification. Primary 55P10. Key words and phrases. Homotopy equivalences, the group of homotopy equivalences, homotopy equivalences which induce the identity, Moore spaces, co-Moore spaces, products of spheres.

Subgroups of the group of self-homotopy equivalences

“…Let X be a Moore space M (G, n), for n ≥ 2 and G any infinite, finitely-generated abelian group with torsion. Then it follows from results of [AM98] that E * (X) = E * fg (X) In Corollary 4.6, we showed that for a 1-connected, finite-dimensional complex X, E * fg (X) is solvable. This raises the question of whether or not the group is nilpotent.…”

“…This kind of example has been considered previously (cf. [AM98,Saw75]), but here we put it into the context discussed above.…”

Self-Homotopy Equivalences Related to Cohomotopy Groups

“…In this section, we review and summarize selected definitions and results provided in [1,3,5,12], knowledge of which would be useful when reading this paper.…”

“…Further, [3] also introduced the forms of the homomorphism induced by f on homotopy, homology, and cohomology groups, respectively. Now, we determine the form of the homomorphism induced by f on cohomotogy groups.…”

“…Then, E(X) is a subset of [X, X] and has a group structure given by the composition of homotopy classes. E(X) has been studied extensively by various authors, including Arkowitz [2], Maruyama [3], Lee [7], Rutter [8], Sawashita [9], and Sieradski [10]. Moreover, several subgroups of E(X) have also been studied, notably the subgroup E k ♯ (X), which consists of the elements of E(X) that induce the identity homomorphism on homotopy 1 1 1…”