On smoothable surgery for 4–manifolds

Khan, Qayum¹

¹Department of Mathematics, Vanderbilt University, Nashville TN 37240, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 4, pp. 2117-2140

Résumé

Under certain homological hypotheses on a compact 4–manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4–manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman–Quinn topological surgery. Necessarily, the ∗–construction of certain non-smoothable homotopy equivalences requires surgery on topologically embedded 2–spheres and is not attacked here by transversality and cobordism.

Détail

DOI : 10.2140/agt.2007.7.2117

Keywords: normal invariants, cobordism

Bibliographie

[1] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)

[2] S E Cappell, Unitary nilpotent groups and Hermitian $K$–theory I, Bull. Amer. Math. Soc. 80 (1974) 1117

[3] S E Cappell, A splitting theorem for manifolds, Invent. Math. 33 (1976) 69

[4] S E Cappell, J L Shaneson, Some new four-manifolds, Ann. of Math. $(2)$ 104 (1976) 61

[5] A Cavicchioli, F Hegenbarth, F Spaggiari, Manifolds with poly-surface fundamental groups, Monatsh. Math. 148 (2006) 181

[6] T D Cochran, N Habegger, On the homotopy theory of simply connected four manifolds, Topology 29 (1990) 419

[7] J F Davis, The Borel/Novikov conjectures and stable diffeomorphisms of 4–manifolds, from: "Geometry and topology of manifolds", Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 63

[8] F T Farrell, L E Jones, Rigidity for aspherical manifolds with $\pi_1{\subset}\mathrm{GL}_m(\mathbb{R})$, Asian J. Math. 2 (1998) 215

[9] M H Freedman, F Quinn, Topology of 4–manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)

[10] M H Freedman, P Teichner, 4–manifold topology I: Subexponential groups, Invent. Math. 122 (1995) 509

[11] I Hambleton, M Kreck, P Teichner, Nonorientable 4–manifolds with fundamental group of order 2, Trans. Amer. Math. Soc. 344 (1994) 649

[12] J A Hillman, On 4–manifolds homotopy equivalent to surface bundles over surfaces, Topology Appl. 40 (1991) 275

[13] J A Hillman, Four-manifolds, geometries and knots, Geom. Topol. Monogr. 5, Geometry Topology Publications, Coventry (2002)

[14] Q Khan, On connected sums of real projective spaces, PhD thesis, Indiana University (2006)

[15] R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann. of Math. Studies 88, Princeton University Press (1977)

[16] R C Kirby, L R Taylor, A survey of 4–manifolds through the eyes of surgery, from: "Surveys on surgery theory, Vol. 2", Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 387

[17] V S Krushkal, R Lee, Surgery on closed 4–manifolds with free fundamental group, Math. Proc. Cambridge Philos. Soc. 133 (2002) 305

[18] V S Krushkal, F Quinn, Subexponential groups in 4–manifold topology, Geom. Topol. 4 (2000)

[19] S López De Medrano, Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 59, Springer (1971)

[20] I Madsen, R J Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies 92, Princeton University Press (1979)

[21] R J Milgram, A A Ranicki, The $L$–theory of Laurent extensions and genus 0 function fields, J. Reine Angew. Math. 406 (1990) 121

[22] A A Ranicki, Algebraic $L$–theory III: Twisted Laurent extensions, from: "Algebraic K–theory, III: Hermitian K–theory and geometric application (Proc. Conf. Seattle Res. Center, Battelle Memorial Inst., 1972)", Lecture Notes in Math. 343, Springer (1973) 412

[23] A Ranicki, The algebraic theory of surgery II: Applications to topology, Proc. London Math. Soc. $(3)$ 40 (1980) 193

[24] A A Ranicki, Algebraic $L$–theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge University Press (1992)

[25] C P Rourke, The Hauptvermutung according to Casson and Sullivan, from: "The Hauptvermutung book", $K$–Monogr. Math. 1, Kluwer Acad. Publ. (1996) 129

[26] S K Roushon, $L$–theory of 3–manifolds with nonvanishing first Betti number, Internat. Math. Res. Notices (2000) 107

[27] S K Roushon, Vanishing structure set of Haken 3–manifolds, Math. Ann. 318 (2000) 609

[28] J L Shaneson, Non-simply-connected surgery and some results in low dimensional topology, Comment. Math. Helv. 45 (1970) 333

[29] D P Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, from: "The Hauptvermutung book", $K$–Monogr. Math. 1, Kluwer Acad. Publ. (1996) 69

[30] L Taylor, B Williams, Surgery spaces: formulae and structure, from: "Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978)", Lecture Notes in Math. 741, Springer (1979) 170

[31] R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17

[32] F Waldhausen, Algebraic $K$–theory of generalized free products I, II, Ann. of Math. $(2)$ 108 (1978) 135

[33] C T C Wall, Classification of Hermitian Forms VI: Group rings, Ann. of Math. $(2)$ 103 (1976) 1

[34] C T C Wall, Surgery on compact manifolds, Mathematical Surveys and Monographs 69, American Mathematical Society (1999)