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 $$\underset{\begin{array}{c}(x,y)\in \mathcal{M}\times \mathcal{M}\\ d_{\mathcal{N}}(f(x),f(y))\ge \epsilon \end{array}}{\iint }\frac{1}{d_{\mathcal{M}}(x,y{)}^{2m}}\mathrm{d}x\mathrm{d}y.$$ When $m\ge 2$, the estimates scale optimally as $\epsilon \to 0$. Linear estimates on the Hurewicz homorphism and the induced cohomology homomorphism are also obtained.

45 pages, minor corrections