Reducibility of self-maps in monoid and its related invariants
Gopal Chandra Dutta
TL;DR
The paper develops criteria for the $k$-reducibility of self-maps on wedge sums in both homology and cohomology settings, linking reducibility to induced endomorphisms and exploiting cohomology-ring structure to test reducibility. It introduces the notion of $n$-atomic spaces and shows that wedges of such spaces yield controlled reducibility behavior, enabling exact computation of homology self-closeness numbers. By leveraging nilpotency, Jacobson radical concepts, and homological distance, the authors derive practical tests ensuring $k$-reducibility under broad conditions. The results culminate in explicit, often maximal, formulas for the homology self-closeness numbers of wedges (and related constructions) in terms of the summands, with concrete examples including Moore/Eilenberg–MacLane spaces and projective spaces, thus facilitating straightforward calculations of self-equivalences in these spaces.
Abstract
Given a positive integer $k$, we investigate the $k$-redcibility of self-maps in the monoid $Å^k(X\vee Y)$, consisting of self-maps that induce isomorphisms on homology groups up to degree $k$. In general, verifying $k$-reducibility is a subtle problem. We show that the $k$-reducibility of a self-map is determine through its induced endomorphisms on homology or cohomology groups. Moreover, under the k-reducibility assumption, the computation of the homology self-closeness number of the wedge sum of spaces essentially reduces to the computation of the homology self-closeness numbers of the individual wedge summands. We generalize the notion of an atomic space to that of an $n$-atomic space and establish some of its fundamental properties. We show that the $k$-reducibility criteria for self-maps in a monoid $Å^k(X)$ is satisfied when the space $X$ decomposes as a wedge sum of distinct $n$-atomic spaces. Finally, we determine the homology self-closeness numbers of wedge sums of distinct $n$-atomic spaces.