On $π_1$-injectivity of self-maps in low dimensions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917132438077440 |
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| author | Neofytidis, Christoforos |
| author_facet | Neofytidis, Christoforos |
| contents | We show that all self-maps of non-zero degree of $3$-manifolds not covered by $S^3$ and of Thurston geometric $4$-manifolds and their connected sums not covered by $N\#(\#_{p\geq0}S^2\times S^2)\#(\#_{q\geq0}\mathbb C P^2)$, where $N$ is an $S^2\times\mathbb X^2$ or $S^3\times\mathbb R$ manifold, are $π_1$-injective. We thus determine when these maps induce $π_1$-isomorphisms. The results in dimension three were previously established by Shicheng Wang. We give a uniform group theoretic proof in all cases based only on the residual finiteness of the fundamental groups for the $π_1$-injectivity and then only on numerical invariants for the $π_1$-isomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $π_1$-injectivity of self-maps in low dimensions Neofytidis, Christoforos Geometric Topology Algebraic Topology Group Theory We show that all self-maps of non-zero degree of $3$-manifolds not covered by $S^3$ and of Thurston geometric $4$-manifolds and their connected sums not covered by $N\#(\#_{p\geq0}S^2\times S^2)\#(\#_{q\geq0}\mathbb C P^2)$, where $N$ is an $S^2\times\mathbb X^2$ or $S^3\times\mathbb R$ manifold, are $π_1$-injective. We thus determine when these maps induce $π_1$-isomorphisms. The results in dimension three were previously established by Shicheng Wang. We give a uniform group theoretic proof in all cases based only on the residual finiteness of the fundamental groups for the $π_1$-injectivity and then only on numerical invariants for the $π_1$-isomorphisms. |
| title | On $π_1$-injectivity of self-maps in low dimensions |
| topic | Geometric Topology Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2512.07238 |