Fundamental groups of reduced suspensions are locally free
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909986857156608 |
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| author | Brazas, Jeremy Gillespie, Patrick |
| author_facet | Brazas, Jeremy Gillespie, Patrick |
| contents | In this paper, we analyze the fundamental group $π_1(ΣX,\overline{x_0})$ of the reduced suspension $ΣX$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $π_1(ΣX,\overline{x_0})$ is canonically isomorphic to a direct limit $\varinjlim_{A\in\mathscr{P}}π_1(ΣA,\overline{x_0})$ where each group $π_1(ΣA,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $π_1(ΣX,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $ΣX$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10499 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fundamental groups of reduced suspensions are locally free Brazas, Jeremy Gillespie, Patrick Algebraic Topology 55Q52, 08A65 In this paper, we analyze the fundamental group $π_1(ΣX,\overline{x_0})$ of the reduced suspension $ΣX$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $π_1(ΣX,\overline{x_0})$ is canonically isomorphic to a direct limit $\varinjlim_{A\in\mathscr{P}}π_1(ΣA,\overline{x_0})$ where each group $π_1(ΣA,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $π_1(ΣX,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $ΣX$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$. |
| title | Fundamental groups of reduced suspensions are locally free |
| topic | Algebraic Topology 55Q52, 08A65 |
| url | https://arxiv.org/abs/2211.10499 |