Coincidence of maps on torus bundle over the circle

• Grant number: 11/12984-0
• Support Opportunities: Regular Research Grants
• Start date: October 01, 2011
• End date: September 30, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: João Peres Vieira
• Grantee: João Peres Vieira
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

Let $T$ be the torus and let $T\to M\stackrel{p}{\to}S^{1}$ be a torus bundle over $S^{1}$. In this work we intend to study the following question: given a pair of fiber preserving maps over $S^1$ when can it be deformed by a fiberwisehomotopy over $S^1$ into a pair of coincidence free fiber preserving maps over $S^1$? Answering this question is equivalent to study existence of a section in a geometric diagram or equivalently, to study the existence of a lifting involving thefundamental groups of the spaces $M$, $M\times_{S^1} M$ and $M\times_{S^1} M-\Delta$, where $M\times_{S^1}M$ is the pullback of $p:M\to S^1$ by $p:M\to S^1$ and $\Delta$ is the diagonal in $M\times_{S^1}M$. We intend to classify all the pairs of maps $(f,g)$ which can be deformed, by fiberwise homotopy over $S^1$, to a pair of maps $(f^{'},g^{'})$ coincidence free. (AU)