Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

Abstract

Abstract A free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in Ws,p(𝓜, 𝓝), with sp = m = dim 𝓜. In particular, when the fundamental group π1(𝓝) acts trivially on the homotopy group πm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$ When m ≥ 2, the estimates scale optimally as ε → 0. When m = 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.