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

Abstract

A free homotopy decomposition of any continuous map from a compact Riemmanian manifold $\mathcal{M}$ to a compact Riemannian manifold $\mathcal{N}$ 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 $W^{s,p} (\mathcal{M}, \mathcal{N})$, with $sp = m = \dim \mathcal{M}$. In particular, when the fundamental group $\pi_1 (\mathcal{N})$ acts trivially on the homotopy group $\pi_m (\mathcal{N})$, 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 $$\iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}}\frac{1}{d_\mathcal{M} (x, y)^{2 m}} \, \mathrm{d} x \, \mathrm{d} y.$$ When $m \ge 2$, the estimates scale optimally as $\varepsilon \to 0$. Linear estimates on the Hurewicz homorphism and the induced cohomology homomorphism are also obtained.