Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T08:53:44.637Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Classification of Manifolds Up to Finite Ambiguity

Published online by Cambridge University Press:  20 November 2018

Matthias Kreck  and
Georgia Triantafillou
Matthias Kreck
Affiliation:
Fachbereich Mathematik University Mainz Mainz, West Germany
Georgia Triantafillou
Affiliation:
Department of Mathematics University of Crete Crete, Greece
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the early 70's Dennis Sullivan applied his theory of minimal models and surgery to the classification of 1-connected closed smooth manifolds of dimension ≥ 5 up to finite ambiguity [Su]. To a diffeomorphism class of such a manifold M he assigns the isomorphism class given by the real minimal model ℳ (M), the integral structure in form of various lattices and the real Pontryagin classes. If one controls the torsion of the manifolds by some bound, his result is that the map given by the triple above is finite-to-one ([Su], Theorem 13.1). He also proves a realization result for the rational minimal model and the Pontryagin classes but not for the lattices ([Su], Theorem 13.2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 2 , 01 April 1991 , pp. 356 - 370
Copyright
Copyright © Canadian Mathematical Society 1991

References

[DGMS] Deligne, P., Griffith, P., Morgan, J., Sullivan, D., The real homotopy of Kaehler manifolds, Invent. Math. 29(1975), 245–274.Google Scholar
[DW] Dold, A., Whitney, H., Classification of oriented sphere bundles over a 4-complex, Ann. of Math (2) 69(1959), 667–677.Google Scholar
[HS] Halperin, S., Stasheff, J., Obstructions to homotopy equivalences, Advances in Math. 32(1979), 233–279.Google Scholar
[Hi] Hirzebruch, F., Topological methods in algebraic geometry. 3rd éd., Springer-Verlag, 1966.Google Scholar
[Kr] Kreck, M., An extension of results of Browder, Novikov and Wall about surgery on compact manifolds, preprint, Mainz (1985), to appear in Vieweg Verlag.Google Scholar
[Mi] Miller, T., On the formality of (k — 1)-connected compact manifolds of dimension less or equal to 4k — 2, Illinois J. Math 23(1979), 253–258.Google Scholar
[Mil] Milnor, J., On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64(1956), 399–405.Google Scholar
[Se] Serre, J.P., Groups d'homotopie et classes des groupes abeliens, Ann. of Math. 58(1953), 258–294.Google Scholar
[St] Stong, R.E., Relations among characteristic numbers I, Topology 4(1965), 267-281; II, ibid, 5(1966), 133–148.Google Scholar
[Su] Sullivan, D., Infinitesimal computations in topology, Publ. Math. IHES 47(1977), 269–331.Google Scholar