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A free homotopy decomposition of any continuous map from a compact Riemmanianmanifold $\mathcal{M}$ to a compact Riemannian manifold $\mathcal{N}$ into afinite number maps belonging to a finite set is constructed, in such a way thatthe number of maps in this free homotopy decomposition and the number ofelements of the set to which they belong can be estimated a priori by thecritical 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 beestimated by its Sobolev energy. The estimates are particular cases ofestimates under a boundedness assumption on gap potentials of the form$$\iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}}\frac{1}{d_\mathcal{M} (x, y)^{2m}} \, \mathrm{d} x \, \mathrm{d} y.$$ When $m \ge 2$, the estimates scaleoptimally as $\varepsilon \to 0$. Linear estimates on the Hurewicz homorphismand the induced cohomology homomorphism are also obtained.

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