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Limiting cases of Boardman's five halves theorem

Part of: Differential topology

Published online by Cambridge University Press:  28 June 2013

Michael C. Crabb  and
Pedro L. Q. Pergher
Michael C. Crabb
Affiliation:
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK ([email protected])
Pedro L. Q. Pergher
Affiliation:
Departamento de Matemática, Universidade Federal de Säo Carlos, Caixa Postal 676, Säo Carlos, SP 13565-905, Brazil ([email protected])
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Abstract

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The famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and if

is the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

Keywords

five halves theoreminvolutionfixed-point dataequivariant cobordism classStiefel–Whitney classcharacteristic number

MSC classification

Secondary: 57R85: Equivariant cobordism 57R75: ${rm O}$- and ${rm SO}$-cobordism
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 723 - 732
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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