Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles
Given a simply connected manifold M, we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M-bundles over the k-sphere, provided that k is small compared to the dimension and the connectivity of M. Furthermore, we study the vector space of rational...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100546/type/journal_article |
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| author | Georg Frenck Jens Reinhold |
| author_facet | Georg Frenck Jens Reinhold |
| author_sort | Georg Frenck |
| collection | DOAJ |
| description | Given a simply connected manifold M, we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M-bundles over the k-sphere, provided that k is small compared to the dimension and the connectivity of M. Furthermore, we study the vector space of rational cobordism classes represented by such bundles. We give upper and lower bounds on its dimension, and we construct manifolds for which the lower bound is attained. Our proofs are based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory, and we make use of ideas developed by Krannich–Kupers–Randal-Williams. |
| format | Article |
| id | doaj-art-cee33d9bb4994329bec84c9adbf7e7fe |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-cee33d9bb4994329bec84c9adbf7e7fe2025-08-26T12:15:03ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10054Characteristic numbers of manifold bundles over spheres and positive curvature via block bundlesGeorg Frenck0https://orcid.org/0000-0002-4260-7797Jens Reinhold1Institut für Mathematik, https://ror.org/03p14d497Universität Augsburg, Universitätsstraße 14, Augsburg, 86159, Bavaria, Germany,Institut für Mathematik, https://ror.org/03p14d497Universität Augsburg, Universitätsstraße 14, Augsburg, 86159, Bavaria, Germany,Given a simply connected manifold M, we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M-bundles over the k-sphere, provided that k is small compared to the dimension and the connectivity of M. Furthermore, we study the vector space of rational cobordism classes represented by such bundles. We give upper and lower bounds on its dimension, and we construct manifolds for which the lower bound is attained. Our proofs are based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory, and we make use of ideas developed by Krannich–Kupers–Randal-Williams.https://www.cambridge.org/core/product/identifier/S2050509425100546/type/journal_article57R2057R2253C2155R4058D1758D05 |
| spellingShingle | Georg Frenck Jens Reinhold Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles Forum of Mathematics, Sigma 57R20 57R22 53C21 55R40 58D17 58D05 |
| title | Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| title_full | Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| title_fullStr | Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| title_full_unstemmed | Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| title_short | Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| title_sort | characteristic numbers of manifold bundles over spheres and positive curvature via block bundles |
| topic | 57R20 57R22 53C21 55R40 58D17 58D05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100546/type/journal_article |
| work_keys_str_mv | AT georgfrenck characteristicnumbersofmanifoldbundlesoverspheresandpositivecurvatureviablockbundles AT jensreinhold characteristicnumbersofmanifoldbundlesoverspheresandpositivecurvatureviablockbundles |