Tautological classes and higher signatures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910786395308032 |
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| author | Ebert, Johannes |
| author_facet | Ebert, Johannes |
| contents | For a bundle of oriented closed smooth $n$-manifolds $π: E \to X$, the tautological class $κ_{\mathcal{L}_k} (E) \in H^{4k-n}(X;\mathbb{Q})$ is defined by fibre integration of the Hirzebruch class $\mathcal{L}_k (T_v E)$ of the vertical tangent bundle. More generally, given a discrete group $G$, a class $u \in H^p(B G;\mathbb{Q})$ and a map $f:E \to B G$, one has tautological classes $κ_{\mathcal{L}_k ,u}(E,f) \in H^{4k+p-n}(X;\mathbb{Q})$ associated to the Novikov higher signatures.
For odd $n$, it is well-known that $κ_{\mathcal{L}_k}(E)=0$ for all bundles with $n$-dimensional fibres. The aim of this note is to show that the question whether more generally $κ_{\mathcal{L}_k,u}(E,f)=0$ (for odd $n$) depends sensitively on the group $G$ and the class $u$.
For example, given a nonzero cohomology class $u \in H^2 (B π_1 (Σ_g);\mathbb{Q})$ of a surface group, we show that always $κ_{\mathcal{L}_k,u}(E,f)=0$ if $g \geq 2$, whereas sometimes $κ_{\mathcal{L}_k,u}(E,f)\neq 0$ if $g=1$.
The vanishing theorem is obtained by a generalization of the index-theoretic proof that $κ_{\mathcal{L}_k}(E)=0$, while the nontriviality theorem follows with little effort from the work of Galatius and Randal-Williams on diffeomorphism groups of even-dimensional manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tautological classes and higher signatures Ebert, Johannes Geometric Topology Algebraic Topology For a bundle of oriented closed smooth $n$-manifolds $π: E \to X$, the tautological class $κ_{\mathcal{L}_k} (E) \in H^{4k-n}(X;\mathbb{Q})$ is defined by fibre integration of the Hirzebruch class $\mathcal{L}_k (T_v E)$ of the vertical tangent bundle. More generally, given a discrete group $G$, a class $u \in H^p(B G;\mathbb{Q})$ and a map $f:E \to B G$, one has tautological classes $κ_{\mathcal{L}_k ,u}(E,f) \in H^{4k+p-n}(X;\mathbb{Q})$ associated to the Novikov higher signatures. For odd $n$, it is well-known that $κ_{\mathcal{L}_k}(E)=0$ for all bundles with $n$-dimensional fibres. The aim of this note is to show that the question whether more generally $κ_{\mathcal{L}_k,u}(E,f)=0$ (for odd $n$) depends sensitively on the group $G$ and the class $u$. For example, given a nonzero cohomology class $u \in H^2 (B π_1 (Σ_g);\mathbb{Q})$ of a surface group, we show that always $κ_{\mathcal{L}_k,u}(E,f)=0$ if $g \geq 2$, whereas sometimes $κ_{\mathcal{L}_k,u}(E,f)\neq 0$ if $g=1$. The vanishing theorem is obtained by a generalization of the index-theoretic proof that $κ_{\mathcal{L}_k}(E)=0$, while the nontriviality theorem follows with little effort from the work of Galatius and Randal-Williams on diffeomorphism groups of even-dimensional manifolds. |
| title | Tautological classes and higher signatures |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2403.02755 |