Rational characteristic classes of bundles with fibre a product of spheres
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909004794429440 |
|---|---|
| author | McGarry-Furriol, Jan |
| author_facet | McGarry-Furriol, Jan |
| contents | We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $ S^{n}\times S^{n} $ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $ S^{n}\times S^{n} $; the latter is proven by Berglund--Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $ Γ$ of $ \mathrm{SL}_{2}(\mathbb{Z}) $. These characteristic classes of smooth bundles are not generalised Miller--Morita--Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $ S^{n}\times S^{n} $-bundles, indexed by cyclic subgroups of $ Γ$, which detect any given non-zero characteristic class of such fibrations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27948 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational characteristic classes of bundles with fibre a product of spheres McGarry-Furriol, Jan Algebraic Topology 55R40 (Primary), 55P62, 55R10, 57R20 (Secondary) We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $ S^{n}\times S^{n} $ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $ S^{n}\times S^{n} $; the latter is proven by Berglund--Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $ Γ$ of $ \mathrm{SL}_{2}(\mathbb{Z}) $. These characteristic classes of smooth bundles are not generalised Miller--Morita--Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $ S^{n}\times S^{n} $-bundles, indexed by cyclic subgroups of $ Γ$, which detect any given non-zero characteristic class of such fibrations. |
| title | Rational characteristic classes of bundles with fibre a product of spheres |
| topic | Algebraic Topology 55R40 (Primary), 55P62, 55R10, 57R20 (Secondary) |
| url | https://arxiv.org/abs/2604.27948 |