Homotopically Non-trivial Maps with Small K-dilation

We construct homotopically non-trivial maps from Sm to Sn with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is the case m = 4, n = 3, and there are other pairs with arbitrarily large values of both m and n. We show that a homotopy class in π7(S) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion. The k-dilation of a map measures how much the map stretches k-dimensional volumes. If f is a C map between Riemannian manifolds, we say that the kdilation of f is at most D if f maps each k-dimensional submanifold of the domain with volume V to an image with volume at most DV . We get the same k-dilation whether we consider all submanifolds or whether we consider only small disks, and so the k-dilation can also be defined in terms of the first derivative df. Recall that Λdf , the k-fold exterior product of the derivative df , maps ΛTM to ΛTN . If f is C, the k-dilation of f is equal to the supremal value of the norm |Λdf |. In this paper, we examine to what extent a bound on the k-dilation of a map controls the homotopy type of the map. A beautiful result of this kind was recently obtained by Tsui and Wang in [7]. Theorem. (Tsui and Wang) Let f be a C map from S to S, where m ≥ 2. If the 2-dilation of f is less than 1, then f is nullhomotopic. The main result of this paper shows that the situation is very different for 3dilation. Theorem 1. For each n, there are infinitely many m so that the following holds: there are homotopically non-trivial maps from S to S with arbitrarily small 3dilation. This result partly answers a question raised by Gromov in [5] (page 231). Gromov asked for which values of k, q, m, and n is a map f : S → S with a sufficiently small norm |Λdf |Lq necessarily null-homotopic. We make the following definition. A homotopy class a in πm(S ) lies in Vkπm(S ) if there are maps in the homotopy class a with arbitrarily small k-dilation. We will prove that Vkπm(S ) is a subgroup of πm(S ) and that Vkπm(S ) ⊂ Vk+1πm(S ). Therefore, Vkπm(S ) defines a filtration of πm(S ). More generally, we will define a filtration Vkπm(X) for any space X. Our methods give some partial information about the filtration Vkπm(S ). The information is most interesting for the filtration Vkπ7(S ). Recall that π7(S ) is isomorphic to Z ⊕ Z12. Theorem 2. The group V4π7(S ) is the torsion subgroup of π7(S ). It is a proper, non-zero subgroup.