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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2312.02527 |
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| _version_ | 1866911833732939776 |
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| author | Gu, Shijie |
| author_facet | Gu, Shijie |
| contents | In 1976, Chapman and Siebenmann \cite{CS76} established necessary and sufficient conditions for $\mathcal{Z}$-compactifying Hilbert cube manifolds. While these conditions are known to be necessary for a manifold $M^n$ to admit a $\mathcal{Z}$-compactification, it remains an open question whether these conditions are also sufficient. Guilbault and the author \cite[Thm. 1.2]{GG20} proved that these conditions are sufficient for the product $M^n \times [-2,2]$ $(n\geq 5)$ to be $\mathcal{Z}$-compactifiable. We further explore this topic by introducing additional conditions such that a $\mathcal{Z}$-compactification of $M^n \times [-2,2]$ indeed implies a $\mathcal{Z}$-compactification of $M^n$, thus partially resolving the open question. As applications, it is shown that there exist infinitely many non-pseudo-collarable 4-manifolds which are $\mathcal{Z}$-compactifiable; however, pseudo-collarable manifolds with compact boundary of dimension at least six are $\mathcal{Z}$-compactifiable.
Furthermore, we investigate the connection between $\mathcal{Z}$-compactifiability with the topological rigidity of aspherical manifolds. We also construct a noncompact one-sided $s$-cobordism $(W,V,V^{\ast})$ satisfying controlled Mather-Thurston theorems, where $V$ is $\mathcal{Z}$-compactifiable, whereas $V^{\ast}$ may not be. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02527 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Z-compactifiability of manifolds Gu, Shijie Geometric Topology In 1976, Chapman and Siebenmann \cite{CS76} established necessary and sufficient conditions for $\mathcal{Z}$-compactifying Hilbert cube manifolds. While these conditions are known to be necessary for a manifold $M^n$ to admit a $\mathcal{Z}$-compactification, it remains an open question whether these conditions are also sufficient. Guilbault and the author \cite[Thm. 1.2]{GG20} proved that these conditions are sufficient for the product $M^n \times [-2,2]$ $(n\geq 5)$ to be $\mathcal{Z}$-compactifiable. We further explore this topic by introducing additional conditions such that a $\mathcal{Z}$-compactification of $M^n \times [-2,2]$ indeed implies a $\mathcal{Z}$-compactification of $M^n$, thus partially resolving the open question. As applications, it is shown that there exist infinitely many non-pseudo-collarable 4-manifolds which are $\mathcal{Z}$-compactifiable; however, pseudo-collarable manifolds with compact boundary of dimension at least six are $\mathcal{Z}$-compactifiable. Furthermore, we investigate the connection between $\mathcal{Z}$-compactifiability with the topological rigidity of aspherical manifolds. We also construct a noncompact one-sided $s$-cobordism $(W,V,V^{\ast})$ satisfying controlled Mather-Thurston theorems, where $V$ is $\mathcal{Z}$-compactifiable, whereas $V^{\ast}$ may not be. |
| title | On Z-compactifiability of manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2312.02527 |