Reducibility of self-maps in monoid and its related invariants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914358246768640 |
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| author | Dutta, Gopal Chandra |
| author_facet | Dutta, Gopal Chandra |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22908 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reducibility of self-maps in monoid and its related invariants Dutta, Gopal Chandra Algebraic Topology 55P10, 55Q05, 55P05, 55P30, 55S45 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. |
| title | Reducibility of self-maps in monoid and its related invariants |
| topic | Algebraic Topology 55P10, 55Q05, 55P05, 55P30, 55S45 |
| url | https://arxiv.org/abs/2601.22908 |