Homotopy theoretic properties of open books
Homotopy theoretic properties of open books
We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decompo- sition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.
open book, loop space decomposition, rational dichotomy, monodromy, Milnor open book
Huang, Ruizhi
86809c85-b4f2-4048-9894-c37cd234e12c
Theriault, Stephen
5e442ce4-8941-41b3-95f1-5e7562fdef80
Huang, Ruizhi
86809c85-b4f2-4048-9894-c37cd234e12c
Theriault, Stephen
5e442ce4-8941-41b3-95f1-5e7562fdef80
Huang, Ruizhi and Theriault, Stephen
(2024)
Homotopy theoretic properties of open books.
The Quarterly Journal of Mathematics.
(In Press)
Abstract
We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decompo- sition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.
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openbook
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Restricted to Repository staff only until 13 June 2025.
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Accepted/In Press date: 13 June 2024
Keywords:
open book, loop space decomposition, rational dichotomy, monodromy, Milnor open book
Identifiers
Local EPrints ID: 491614
URI: http://eprints.soton.ac.uk/id/eprint/491614
ISSN: 0033-5606
PURE UUID: 4c2a94c7-9ef9-4ff0-9a63-e1548ce4d241
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Date deposited: 28 Jun 2024 16:35
Last modified: 12 Jul 2024 01:50
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Contributors
Author:
Ruizhi Huang
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