Homotopy counting S 1 ‐ and S 2 ‐valued maps with prescribed dilatation

Let V , W be two compact Riemannian manifolds and # [ V , W ] D the number of homotopy classes of maps with dilatation less than or equal to D . It is shown that ( c 1 D − 1) b ⩽ # [ M , S 1 ] D ⩽ ( c 2 D + 1) b , where b = b 1 ( M ) is the first Betti number of M . The second result is that if M is a closed oriented Riemannian 3‐manifold, then the number of homotopy classes of algebraically trivial maps M → S 2 with dilatation less than D is at most c 3 D 4 . This result covers an earlier theorem by Gromov. Finally, we prove that if M is a closed oriented Riemannian 3‐manifold with H 1 ( M , ℤ) torsion free, then # [ M , S 2 ] D ⩽ c 3 D 4 + c 4 D 2 b +2 . The above constants c i depend on the metrics on the manifolds concerned.