Stable diffeomorphism classification of some unorientable 4-manifolds

Arun Debray

doi:10.1112/blms.12688

Stable diffeomorphism classification of some unorientable 4-manifolds

Arun Debray

Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of (Formula presented.), to a series of bordism questions. We implement this in the case of unorientable 4-manifolds (Formula presented.) and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which (Formula presented.) is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which (Formula presented.) is pin (Formula presented.), one stable diffeomorphism class for which (Formula presented.) is pin (Formula presented.), and four stable diffeomorphism classes for which (Formula presented.) is neither. We also determine the corresponding stable homeomorphism classes.

Original language	English
Pages (from-to)	2219-2231
Number of pages	13
Journal	Bulletin of the London Mathematical Society
Volume	54
Issue number	6
DOIs	https://doi.org/10.1112/blms.12688
State	Published - Dec 2022

Bibliographical note

Publisher Copyright:
© 2022 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.

Funding

A portion of this work was supported by the National Science Foundation under Grant No. 1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during January–March 2020.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1440140
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China

ASJC Scopus subject areas

General Mathematics

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abstract = "Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of (Formula presented.), to a series of bordism questions. We implement this in the case of unorientable 4-manifolds (Formula presented.) and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which (Formula presented.) is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which (Formula presented.) is pin (Formula presented.), one stable diffeomorphism class for which (Formula presented.) is pin (Formula presented.), and four stable diffeomorphism classes for which (Formula presented.) is neither. We also determine the corresponding stable homeomorphism classes.",

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AB - Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of (Formula presented.), to a series of bordism questions. We implement this in the case of unorientable 4-manifolds (Formula presented.) and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which (Formula presented.) is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which (Formula presented.) is pin (Formula presented.), one stable diffeomorphism class for which (Formula presented.) is pin (Formula presented.), and four stable diffeomorphism classes for which (Formula presented.) is neither. We also determine the corresponding stable homeomorphism classes.

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Stable diffeomorphism classification of some unorientable 4-manifolds

Abstract

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