Title: Advances in quantitative homotopy theory
Speaker: Fedor Manin (Ohio State University)
Abstract: If two finite complexes or compact manifolds are homotopy equivalent, then they are Lipschitz homotopy equivalent (in the obvious sense.) Therefore, in this context, Lipschitz homotopy invariants are a natural object of study. More than 20 years ago, Gromov made a number of conjectures about such invariants; today we are finally equipped to tackle some of them. The main technique is a kind of inverse to the well-known results of rational homotopy theory: it is possible to produce maps which are "close" to arbitrary DGA homomorphisms. Combined with some quantitative results about DGAs, this proves a number of results about maps from a finite complex $X$ to a finite simply connected complex $Y$, such as: * The number of homotopy classes in $[X,Y]$ which have an $L$-Lipschitz representative grows at most polynomially in $L$. However, pace Gromov, this is not always exactly a polynomial; in one example the growth is asymptotic to $L^8 \log L$. * Any two homotopic $L$-Lipschitz maps $f, g:X \to Y$ are homotopic via a $C(L+1)^p$-Lipschitz map $X \times I \to Y$, where $C$ and $p$ are constants depending on $X$ and $Y$. Many questions remain open. Some of this is joint work with Shmuel Weinberger.
Seminar URL: https://research.math.osu.edu/topology/
