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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
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatástica, Universidade de São Paulo, Caixa Postal 66281, 05314-970 São Paulo, Brazil, (xlink:href="dlgoncal@ime.usp.br">dlgoncal@ime.usp.br)
Ulrich Koschorke
Affiliation:
Department of Mathematics, Universität Siegen, Emmy-Noether-Campus, 57068 Siegen, Germany, (xlink:href="koschorke@mathematik.uni-siegen.de">koschorke@mathematik.uni-siegen.de)
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="alicekml@rc.unesp.br">alicekml@rc.unesp.br)
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="ozimneto@icmc.usp.br">ozimneto@icmc.usp.br)

Abstract

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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