Geometric Analysis and Topology Seminar

Homotopical Effects of K-Dilation

Speaker: Larry Guth, MIT

Location: Warren Weaver Hall 805

Date: Monday, April 21, 2014, 2 p.m.


The \(k\)-dilation of a map measures how much it stretches \(k\)-dimensional areas. If \(\mathrm{Dil}_k f < L\), then it means that for any \(k\)-dimensional submanifold \(S\) in the domain, \(\mathrm{Vol}_k (f(S))\) is at most \(L \mathrm{Vol}_k(S)\). We discuss how the \(k\)-dilation restricts the homotopy type of a map. Our main theorem concerns maps between unit spheres, from \(S^{m}\) to \(S^{m-1}\). If \(k > (m+1)/2\), then there are homotopically non-trivial maps \(S^m\) to \(S^{m-1}\) with arbitrarily small \(k\)-dilation. If \(k\) is at most \((m+1)/2\), there every homotopically non-trivial map from \(S^m\) to \(S^{m-1}\) has \(k\)-dilation at least \(c(m) > 0\). In this talk, I want to focus on the case \(k\) at most \((m+1)/2\). The non-trivial homotopy type of a map \(S^m\) to \(S^{m-1}\) is detected by a certain Steenrod square. The main issue is how to connect Steenrod squares with quantitative estimates about \(k\)-dimensional volumes. This involves a mix of topology and geometry -- on the geometrical side the tools are related to isoperimetric inequalities/geometric measure theory. (We don't assume familiarity with Steenrod squares.).