Joshua Zhong

Thesis Advisor: Dr. Francis Su (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Justin Lanier (Louisiana State University - Department of Mathematics)

Contact: jozhong@hmc.edu

A combinatorial view of the fixed-point property for projective planes of even dimension

There are several topological existence theorems that have interesting equivalent combinatorial lemmas on labeled triangulations. The most prominent is that the Brouwer fixed-point theorem for n-dimensional disks is equivalent to Sperner's lemma for triangulated subdivisions of n-simplices. However, there are other topological spaces that have the fixed-point property. It can be shown that projective planes of odd dimension have continuous, fixed-point free mappings to themselves, but no such functions exist for ones of even dimension. We seek a combinatorial equivalent that shows the existence of these fixed points, particularly by first creating a corresponding mapping from the 2n-dimensional sphere to itself.