Mathematische Zeitschrift

, Volume 269, Issue 3–4, pp 1189–1199 | Cite as

On the mapping class groups of #r(Sp × Sp) for p = 3, 7

Article

Abstract

For \({M_r := \sharp_r(S^p \times S^p),\,p=3, 7}\), we calculate \({\pi_0{\rm Diff}(M_r)/\Theta_{2p+1}}\) and \({\mathcal{E}(M_r)}\), respectively the group of isotopy classes of orientation preserving diffeomorphisms of Mr modulo isotopy classes with representatives which are the identity outside a 2p-disc and the group of homotopy classes of orientation preserving homotopy equivalences of Mr.

Keywords

Mapping class groups Self-homotopy equivalences 

Mathematics Subject Classification (2000)

57R52 55P10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baues H.J.: On the group of homotopy equivalences of a manifold. Trans. A.M.S. 348(12), 4737–4773 (1996)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Brown K.S.: Cohomology of Groups. Springer, New York (1982)MATHGoogle Scholar
  3. 3.
    Cerf, J.: The pseudo-isotopy theorem for simply connected differentiable manifolds, Manifolds-Amsterdam (1970), 76–82, Lect. Notes Math. 197, Springer, Berlin (1970)Google Scholar
  4. 4.
    Fried D.: Word maps, isotopy and entropy. Trans. A.M.S. 296, 851–859 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Kreck, M.: Isotopy classes of diffeomorphisms of (k − 1)-connected almost-parallelizable 2k-manifolds, Algebraic topology, Aarhus (1978), 643–663, Lect. Notes Math. 763, Springer, Berlin (1979)Google Scholar
  6. 6.
    Krylov, N.A.: Mapping class groups of (k − 1)-connected almost parallelizable 2k-manifolds. PhD thesis, University of Illinois (2002)Google Scholar
  7. 7.
    Krylov N.A.: On the Jacobi group and the mapping class group of S 3 × S 3. Trans. A.M.S. 355, 99–117 (2003)MATHCrossRefGoogle Scholar
  8. 8.
    Levine, J.P.: Lectures on groups of homotopy spheres, Algebraic and geometric topology, New Brunswick, NJ (1983), 62–95, Lect. Notes Math. 1126, Springer, Berlin (1985)Google Scholar
  9. 9.
    Lück, W.: A basic introduction to surgery theory, topology of high-dimensional manifolds, No. 1, 2, Trieste (2001), 1–224, ICTP Lect. Notes, 9, Trieste (2002)Google Scholar
  10. 10.
    Wall C.T.C.: Classification of (n − 1)-connected 2n-manifolds. Ann. Math. 75, 163–189 (1962)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations