The Equivariant L-Class of Pseudomanifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912160671596544 |
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| author | Banagl, Markus |
| author_facet | Banagl, Markus |
| contents | We construct an equivariant L-class for orientation preserving actions of a compact Lie group on a Whitney stratified compact oriented pseudomanifold that satisfies the Witt condition, for example on a compact pure-dimensional complex algebraic variety. The class lies in equivariant rational homology and its restriction to the trivial group is the Goresky-MacPherson L-class. For a smooth action on a manifold, the class is equivariantly Poincaré dual to the Hirzebruch L-class of the Borel homotopy quotient of the tangent bundle. We also provide a product formula under the equivariant Künneth isomorphism. If the group acts freely, then the equivariant L-class identifies with the Goresky-MacPherson L-class of the orbit space. The construction method rests on establishing Whitney (B)-regularity for finite-dimensional compact pseudomanifold approximations to the Borel construction, and on the author's Verdier-Riemann-Roch type formulae for the Goresky-MacPherson L-class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Equivariant L-Class of Pseudomanifolds Banagl, Markus Algebraic Topology 55N33, 57N80, 55N91, 57R20, 57R91, 57S15 We construct an equivariant L-class for orientation preserving actions of a compact Lie group on a Whitney stratified compact oriented pseudomanifold that satisfies the Witt condition, for example on a compact pure-dimensional complex algebraic variety. The class lies in equivariant rational homology and its restriction to the trivial group is the Goresky-MacPherson L-class. For a smooth action on a manifold, the class is equivariantly Poincaré dual to the Hirzebruch L-class of the Borel homotopy quotient of the tangent bundle. We also provide a product formula under the equivariant Künneth isomorphism. If the group acts freely, then the equivariant L-class identifies with the Goresky-MacPherson L-class of the orbit space. The construction method rests on establishing Whitney (B)-regularity for finite-dimensional compact pseudomanifold approximations to the Borel construction, and on the author's Verdier-Riemann-Roch type formulae for the Goresky-MacPherson L-class. |
| title | The Equivariant L-Class of Pseudomanifolds |
| topic | Algebraic Topology 55N33, 57N80, 55N91, 57R20, 57R91, 57S15 |
| url | https://arxiv.org/abs/2412.13768 |