Reducibility of self-maps in monoid and its related invariants

Given a positive integer $k$, we investigate the $k$-redcibility of self-maps in the monoid $\AA^k(X\vee Y)$, 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 $\AA^k(X)$ 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.