Bundle Transfer of L-Homology Orientation Classes for Singular Spaces
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929476171988992 |
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| author | Banagl, Markus |
| author_facet | Banagl, Markus |
| contents | We consider transfer maps on ordinary homology, bordism of singular spaces and homology with coefficients in Ranicki's symmetric L-spectrum, associated to block bundles with closed oriented PL manifold fiber and compact polyhedral base. We prove that if the base polyhedron is a Witt space, for example a pure-dimensional compact complex algebraic variety, then the symmetric L-homology orientation of the base, constructed by Laures, McClure and the author, transfers to the L-homology orientation of the total space. We deduce from this that the Cheeger-Goresky-MacPherson L-class of the base transfers to the product of the L-class of the total space with the cohomological L-class of the stable vertical normal microbundle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05432 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bundle Transfer of L-Homology Orientation Classes for Singular Spaces Banagl, Markus Algebraic Topology 55R12 (Primary) 55N33, 57R20, 57N80, 57Q20, 57Q50 (Secondary) We consider transfer maps on ordinary homology, bordism of singular spaces and homology with coefficients in Ranicki's symmetric L-spectrum, associated to block bundles with closed oriented PL manifold fiber and compact polyhedral base. We prove that if the base polyhedron is a Witt space, for example a pure-dimensional compact complex algebraic variety, then the symmetric L-homology orientation of the base, constructed by Laures, McClure and the author, transfers to the L-homology orientation of the total space. We deduce from this that the Cheeger-Goresky-MacPherson L-class of the base transfers to the product of the L-class of the total space with the cohomological L-class of the stable vertical normal microbundle. |
| title | Bundle Transfer of L-Homology Orientation Classes for Singular Spaces |
| topic | Algebraic Topology 55R12 (Primary) 55N33, 57R20, 57N80, 57Q20, 57Q50 (Secondary) |
| url | https://arxiv.org/abs/2210.05432 |