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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2204.11696 |
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| _version_ | 1866916668621455360 |
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| author | Randal-Williams, Oscar |
| author_facet | Randal-Williams, Oscar |
| contents | We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\tfrac{1}{2}]$-theory using ideas of Sullivan, and finally in symmetric $L$-theory using ideas of Ranicki. Employing recent developments in Grothendieck--Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_11696 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The family signature theorem Randal-Williams, Oscar Algebraic Topology 19G24, 19G38, 55R10, 57R19, 57R20 We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\tfrac{1}{2}]$-theory using ideas of Sullivan, and finally in symmetric $L$-theory using ideas of Ranicki. Employing recent developments in Grothendieck--Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant. |
| title | The family signature theorem |
| topic | Algebraic Topology 19G24, 19G38, 55R10, 57R19, 57R20 |
| url | https://arxiv.org/abs/2204.11696 |