Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912184991219712 |
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| author | Bonciocat, Ciprian Mircea |
| author_facet | Bonciocat, Ciprian Mircea |
| contents | In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the alleged stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $Σ^\infty_+ M$. We further show that other systems of compatible stable normal framings recover Thom spectra $M^E$, for all reduced $KO$-theory classes $E$ on $M$. Our paper also includes a construction of the smooth corner structure on compactified moduli spaces of broken flow lines with free endpoint, a formal construction of Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable normal framing condition to orientability in orthogonal spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory Bonciocat, Ciprian Mircea Algebraic Topology Geometric Topology Symplectic Geometry 55P42 (Primary) 37D15, 57R58, 57R19, 53D05 (Secondary) In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the alleged stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $Σ^\infty_+ M$. We further show that other systems of compatible stable normal framings recover Thom spectra $M^E$, for all reduced $KO$-theory classes $E$ on $M$. Our paper also includes a construction of the smooth corner structure on compactified moduli spaces of broken flow lines with free endpoint, a formal construction of Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable normal framing condition to orientability in orthogonal spectra. |
| title | Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory |
| topic | Algebraic Topology Geometric Topology Symplectic Geometry 55P42 (Primary) 37D15, 57R58, 57R19, 53D05 (Secondary) |
| url | https://arxiv.org/abs/2409.11278 |