Abstract
Thom–Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism ϕ : H n(L;Z) → H n(M;Z) to be realized by a map f : M → L of degree k for closed (n − 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1.
In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Edmonds, A.: Deformation of maps to branched covering in dimension 2. Ann. Math., 110, 113–125 (1979)
Wang, S. C.: Non-zero degree maps between 3-manifolds, Proceedings of International Congress of Mathematicains, Beijing, 2002 Vol. II 457-468, High Education Press, Beijing, 2002
Duan, H. B., Wang, S. C.: The degrees of maps between manifolds. Math. Zeit., 244, 67–89 (2003)
Dubrovin, B. A., Fomenko, A. T., Novikov, S.: Modern Geometry, II GTM, Vol. 104, Springer-Verlag, New York, 1984
Wall, C. T. C.: Classification of (n − 1)-connected 2n-manifolds. Ann. of Math., 75, 163–198 (1962)
Hu, S.: Homotopy theory, Academic Press, New York and London, 1959
Kirby, R.: The topology of 4-manifolds, Lecture Notes in Math, Vol. 1374, Springer-Verlag, Berlin, 1989
Milnor, J., Husemoller, D.: Symmetric bilinear forms, Springer-Verlag, Berlin and New York, 1973
Freedman, M. H.: The topology of 4-dimensional manifolds. J. Diff. Geom., 17, 357–453 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Both authors are supported by MSTC, NSFC. The comments of F. Ding, J. Z. Pan, Y. Su and the referee enhance the quality of the paper
Rights and permissions
About this article
Cite this article
Duan, H.B., Wang, S.C. Non-Zero Degree Maps Between 2n-Manifolds. Acta Math Sinica 20, 1–14 (2004). https://doi.org/10.1007/s10114-003-0307-x
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10114-003-0307-x