Session 35. Topological fixed point theory and related topics

The fixed points of multimaps on surface with application to the torus- a Braid approach

Daciberg Lima Gonçalves, Department of Mathematics-Univeristy of São Paulo, Brazil
The talk is based on the joint work with John Guaschi
Let \(\phi:S \to S\) be a n-valued continuous multimap on some compact surface \(S\). First we classify the homotopy classes of multimaps where for most of the surfaces the classification is given in terms of the braids on \(n-\)strings of the surface \(S\). Then we give an algebraic criterion to decide which homotopy classes contains a multimap which is fixed point free. We will focus on the cases where \(S\) is a closed surface of Euler characteristic \(\leq 0\). Despite the fact that the algebraic criterion is quite hard, we performe some specific calculations for the case where \(S\) is the torus. The concept of Nielsen number for a surface has been developed. Then I explain the status of the Wecken property for multimaps on the torus. In fact it is an open question if there is an example of a multimap which has Nielsen number zero but it can not be deformed to fixed point free. Finally a brief exposition about the case of the projective plane should be presented. Below are sme of the relevant references for our purpose.
References
  1. J. Better, A Wecken theorem for \(n-\)valued maps , Top. Applic., 159, 2012, 3707-3715.
  2. Robert F. Brown, Fixed points of \(n\)-valued multimaps of the circle , Bull. Pol. Acad. Sci. Math., 54 no. 2, 2006, 153-162.
  3. Robert F. Brown, The Lefschetz number of an \(n\)-valued multimap , JP J. Fixed Point Theory Appl., 2 no. 1, 2007, 53-60.
  4. Robert F. Brown, Nielsen numbers of \(n\)-valued fiber maps , J. Fixed Point Theory Appl., 4 no. 2, 2008, 183-201.
  5. Robert F. Brown, Lo Kim Lin, Jon T., Coincidences of projections and linear \(n\)-valued maps of tori , Topology Appl., 157 no. 12, 2010, 1990-1998.
  6. D. L. Gonçalves and J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, Topology and its Applications, 157, 2010, 1742-1759.
  7. H. Schirmer, An index and a Nielsen number for \(n-\)valued multifunctions , Fund. Math., 124, 1984, 207-219.
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