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Bulletin of the London Mathematical Society Advance Access originally published online on January 24, 2009
Bulletin of the London Mathematical Society 2009 41(1):124-136; doi:10.1112/blms/bdn113
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© 2009 London Mathematical Society

Homotopy counting S1- and S2-valued maps with prescribed dilatation

Luofei Liu

College of Mathematics and Computer Science
Jishou University
Jishou
Hunan 416000
China

Received 6 November 2007.

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 (c1D – 1)b ≤ # [M, S1]D ≤ (c2D + 1)b, where b = b1(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 -> S2 with dilatation less than D is at most c3D4. 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, S2]D ≤ c3D4 + c4D2b+2. The above constants ci depend on the metrics on the manifolds concerned.

2000 Mathematics Subject Classification 53C23, 53C20.

This work was partially supported by National Natural Science Foundation of China grant no. 10871081 and the Natural Science Foundation of Hunan Province, China grant no. 06JJ5009.


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