Moving cubes around

Dale Rolfsen (University of British Columbia)

Wednesday 22nd April, 2015 16:00-17:00 Maths 516

Abstract

It is an easy exercise to prove that any homeomorphism h of the interval I = [0, 1] which is fixed on the boundary has infinite order.  A 1920 result of Kerekjarto implies a similar result for the square I² (though his proof had a gap, filled later by Eilenberg).  P. A. Smith theory implies the group of homeomorphisms of the cube Iⁿ, fixed on the boundary, is torsion-free, for all positive integers n. 

If one assumes the homeomorphisms are piecewise-linear or smooth, more can be said.  The group of PL (or smooth) homeomorphisms of the n-cube fixed on the boundary has the stronger property of being left-orderable.  This result generalizes to arbitrary compact connected PL or smooth manifolds with boundary.  This is joint work with Danny Calegari and exemplifies the slogan ``think globally, act locally.''

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