Higher Lefschetz formulas on Γ-proper manifolds
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911322648608768 |
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| author | Piazza, Paolo Posthuma, Hessel Song, Yanli Tang, Xiang |
| author_facet | Piazza, Paolo Posthuma, Hessel Song, Yanli Tang, Xiang |
| contents | Let $Γ$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $Γ$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $τ$ in $HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$. Our formula takes place on the fixed point manifold $M^γ$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_γ$-invariant form on $M^γ$ that is naturally associated to $τ\in HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14318 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Lefschetz formulas on Γ-proper manifolds Piazza, Paolo Posthuma, Hessel Song, Yanli Tang, Xiang Differential Geometry K-Theory and Homology 58J20 Let $Γ$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $Γ$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $τ$ in $HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$. Our formula takes place on the fixed point manifold $M^γ$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_γ$-invariant form on $M^γ$ that is naturally associated to $τ\in HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$ |
| title | Higher Lefschetz formulas on Γ-proper manifolds |
| topic | Differential Geometry K-Theory and Homology 58J20 |
| url | https://arxiv.org/abs/2512.14318 |