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Homotopy counting S 1 ‐ and S 2 ‐valued maps with prescribed dilatation
Author(s) -
Liu Luofei
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn113
Subject(s) - mathematics , betti number , homotopy , riemannian manifold , torsion (gastropod) , manifold (fluid mechanics) , homotopy group , combinatorics , pure mathematics , medicine , mechanical engineering , surgery , engineering
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.