Simply connected manifolds with large homotopy stable classes

Conway, A., Crowley, D., Powell, M. and Sixt, J. (2022) Simply connected manifolds with large homotopy stable classes. Journal of the Australian Mathematical Society, (doi: 10.1017/S1446788722000167) (Early Online Publication)

Abstract

For every k ≥ 2 and n ≥ 2, we construct n pairwise homotopically inequivalent simply connected, closed 4k-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension four, we exhibit an analogous phenomenon for spinc structures on S2 × S2. For m ≥ 1, we also provide similar (4m − 1)-connected 8m-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism π4m−1(SO) → πs 4m−1.

• Item Type: Articles
• Additional Information: The third author was partially supported by EPSRC New Investigator grant EP/T028335/1 and EPSRC New Horizons grant EP/V04821X/1.
• Status: Early Online Publication
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Powell, Professor Mark
• Authors: Conway, A., Crowley, D., Powell, M., and Sixt, J.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of the Australian Mathematical Society
• Publisher: Cambridge University Press
• ISSN: 1446-7887
• ISSN (Online): 1446-8107
• Published Online: 26 September 2022
• Copyright Holders: Copyright © 2022 The Authors
• First Published: First published in Journal of the Australian Mathematical Society 2022
• Publisher Policy: Reproduced under a Creative Commons License