Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917413698666496 |
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| author | Banagl, Markus |
| author_facet | Banagl, Markus |
| contents | If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces Banagl, Markus Algebraic Topology K-Theory and Homology 55N33, 57N80, 57R20, 57R91, 57S17 If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds. |
| title | Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces |
| topic | Algebraic Topology K-Theory and Homology 55N33, 57N80, 57R20, 57R91, 57S17 |
| url | https://arxiv.org/abs/2604.14913 |