Let V n be a compact, oriented Riemannian manifold and S n the standard sphere. We study the problem of obtaining upper bounds for the dilatation invariants of maps V n → S n of nonzero degree. The dilatation upper bounds are then used to estimate Gromov's filling volume from below.
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 thatThe 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 c3D 4 . This result covers an earlier theorem by Gromov. Finally, we prove that if M is a closed oriented Riemannian 3-manifold with H1(M, Z) torsion free, then #[M, S 2 ]D c3D 4 + c4D 2b+2 . The above constants ci depend on the metrics on the manifolds concerned.
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