Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Estimates by gap potentials of free homotopy decompositions of critical
  Sobolev maps   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.

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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: physics