Recent Progress in Quantitative Topology

We discuss recent work of Chambers, Dotterrer, Ferry, Manin, and Weinberger, which resolved a fundamental question in quantitative topology: if f : Sm → Sn is a contractible map with Lipschitz constant L, what can we say about the Lipschitz constant of a null-homotopy of f? In the mid-90s, Gromov wrote an article on quantitative topology [G], raising a number of interesting questions. We will focus on the following question. Question 0.1. Equip S and S with the unit sphere metrics, and suppose that f : S → S is a contractible map with Lipschitz constant L. What is the best Lipschitz constant of a null-homotopy H : S × [0, 1]→ S? For a long time, little was known about this question. In some simple cases, there were good estimates. For general m and n, Gromov sketched a construction giving a homotopy H with Lipschitz constant at most exp(exp(...(expL)...)), where the height of the tower of exponentials depends on the dimensions m and n [G]. On the other hand, he suggested that there may always be a homotopy with Lipschitz constant at most C(m,n)L. I thought about the problem a number of times but couldn’t see any way to get started. Recently, the problem was almost completely solved by work of Chambers, Dotterrer, Ferry, Manin, and Weinberger. Here are their results. Theorem 0.2. ([CDMW]) Suppose that n is odd and f : S → S is a contractible map with Lipschitz constant L. Then there is a null-homotopy H : S × [0, 1]→ S with Lipschitz constant at most C(m,n)L. Theorem 0.3. ([CMW]) Suppose that n is even and f : S → S is a contractible map with Lipschitz constant L. Then there is a null-homotopy H : S × [0, 1]→ S with Lipschitz constant at most C(m,n)L. More precisely, there is a null-homotopy H which is C(m,n)L-Lipschitz in the S-direction and C(m,n)L-Lipschitz in the [0, 1]-direction. In other words: distSn(H(x1, t1), H(x2, t2)) ≤ C(m,n) ( LdistSm(x1, x2) + L |t1 − t2| ) . When n is even, it is still an open question whether there is a null-homotopy H with Lipschitz constant at most C(m,n)L. Nevertheless, these results are a huge improvement over what was known before. They give a near complete solution to a fundamental problem. Question 0.1 lies on the border between algebraic topology and metric geometry. It is an inventive variation on the theme of the isoperimetric inequality in the domain of homotopy theory. The proof involves ideas from both areas. On the homotopy theory side, it uses Serre’s classification of the