# Fixed points of multimaps on surfaces with application to the torus- a Braid approach

Daciberg Lima Gonçalves

http://www.ime.usp.br/~dlgoncal/

Date(s) : 20/10/2014   iCal
14 h 00 min - 15 h 00 min

Let $\phi: S \to S$ be an $n-$valued continuous multimap on some closed surface $S$. First we define the set of those maps called split. Then we describe the set of homotopy classes of such multimaps where for most of the surfaces the classification is given in terms of braids on $n-$strings and the pure $n-$braids. The case where $S$ is either $S^2$ or $RP^2$(projective plane) will be explained separately. For the case where the surface has genus > 0 then we give an algebraic criterion to decide which homotopy classes of maps contains a representative which is fixed point free. Despite the fact that the algebraic condition is quite hard, we perform some explicit calculation for the case where $S$ is the torus and explain the sate of art of the problem in this case.

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