Decomposing Four-Manifolds up to Homotopy Type
نویسندگان
چکیده
Let M be a closed connected oriented topological 4-manifold with fundamental group π1. Let Λ be the integral group ring of π1. Suppose that f : M → P is a degree one map inducing an isomorphism on π1. We give a homological condition on the intersection forms λM and λ Λ M under which M is homotopy equivalent to a connected sum P#M ′ for some simply-connected closed (non-trivial) topological 4-manifoldM . This gives a partial solution to a conjecture of Hillman [16] on the classification of closed 4-manifolds with vanishing second homotopy group. Then some splitting results for closed 4-manifolds with special homotopy complete the paper. MSC 2000: 57N65, 57R67, 57Q10
منابع مشابه
On Rational Homotopy of Four-manifolds
We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.
متن کاملManifolds homotopy equivalent to Pn # Pn
We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. 1. Statement of results Let P = Pn(R) be real projective n-space. López de Medrano [LdM71] and C.T.C. Wall [Wal68, Wal99] classified, up to PL homeomorphism, all closed PL manifolds homotopy equivalent to P when n > 4. This was extended to the topological ...
متن کاملJeremy Brookman
We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. 1. Statement of results Let P = Pn(R) be real projective n-space. López de Medrano [LdM71] and C.T.C. Wall [Wal68, Wal99] classified, up to PL homeomorphism, all closed PL manifolds homotopy equivalent to P when n > 4. This was extended to the topological ...
متن کاملSeiberg – Witten – Floer stable homotopy type of three - manifolds with b 1 = 0
Using Furuta’s idea of finite dimensional approximation in Seiberg–Witten theory, we refine Seiberg–Witten Floer homology to obtain an invariant of homology 3–spheres which lives in the S1–equivariant graded suspension category. In particular, this gives a construction of Seiberg–Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a re...
متن کاملSe p 20 03 SEIBERG - WITTEN - FLOER STABLE HOMOTOPY TYPE OF THREE - MANIFOLDS WITH b 1 = 0
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S 1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversal-ity problems in the standard approach. We also define a ...
متن کامل