Homotopy of blow ups after looping
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910993416716288 |
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| author | Huang, Ruizhi Theriault, Stephen |
| author_facet | Huang, Ruizhi Theriault, Stephen |
| contents | The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized by the homotopy groups of a certain gauge group. This is used to obtain a homotopy decomposition of the based loop space on a blow up that holds $p$-locally for all but finitely many primes $p$ and holds rationally. The second approach is purely homotopy theoretic and obtains a homotopy decomposition of the based loop space on a blow up that holds integrally provided a certain condition is satisfied by an associated homotopy action.
As applications, we obtain $p$-local homotopy decompositions of the based loop space of a focal genus $2$ manifold, improve an earlier result of the authors on the homotopy of manifolds stabilized by projective spaces, and obtain for blow-ups a refinement of the rational dichotomy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14531 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homotopy of blow ups after looping Huang, Ruizhi Theriault, Stephen Algebraic Topology Geometric Topology The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized by the homotopy groups of a certain gauge group. This is used to obtain a homotopy decomposition of the based loop space on a blow up that holds $p$-locally for all but finitely many primes $p$ and holds rationally. The second approach is purely homotopy theoretic and obtains a homotopy decomposition of the based loop space on a blow up that holds integrally provided a certain condition is satisfied by an associated homotopy action. As applications, we obtain $p$-local homotopy decompositions of the based loop space of a focal genus $2$ manifold, improve an earlier result of the authors on the homotopy of manifolds stabilized by projective spaces, and obtain for blow-ups a refinement of the rational dichotomy. |
| title | Homotopy of blow ups after looping |
| topic | Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2308.14531 |