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Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

Published online by Cambridge University Press:  22 November 2013

Daciberg L. Gonçalves ,
Ulrich Koschorke ,
Alice K. M. Libardi  and
Oziride Manzoli Neto
Daciberg L. Gonçalves
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatástica, Universidade de São Paulo, Caixa Postal 66281, 05314-970 São Paulo, Brazil, (xlink:href="[email protected]">[email protected])
Ulrich Koschorke
Affiliation:
Department of Mathematics, Universität Siegen, Emmy-Noether-Campus, 57068 Siegen, Germany, (xlink:href="[email protected]">[email protected])
Alice K. M. Libardi
Affiliation:
Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista Júlio de Mesquita Filho, Caixa Postal 178 Rio Claro, São Paulo, Brazil, (xlink:href="[email protected]">[email protected])
Oziride Manzoli Neto
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668 São Carlos, São Paulo, Brazil, (xlink:href="[email protected]">[email protected])
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Abstract

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When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.

In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

Keywords

fibrewise map and homotopycoincidenceNielsen numbersphere bundlenormal bordismGysin sequencePrimary 55M20Secondary 55Q4055R2555S3557R90
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 713 - 735
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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