A note on the complexity of h–cobordisms

Schwartz, Hannah R¹

¹Max Planck Institute of Mathematics, Bonn, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 7, pp. 3313-3327

Résumé

We show that the number of double points of smoothly immersed 2–spheres representing certain homology classes of an oriented, smooth, closed, simply connected 4–manifold X must increase with the complexity of corresponding h–cobordisms from X to X. As an application, we give results restricting the minimal number of double points of immersed spheres in manifolds homeomorphic to rational surfaces.

Détail

DOI : 10.2140/agt.2020.20.3313

Classification : 57Q20, 57Q99
Keywords: topology, manifold, smooth structures, spheres, $h$–cobordism

Bibliographie

[1] S Akbulut, A fake compact contractible 4–manifold, J. Differential Geom. 33 (1991) 335 | DOI

[2] J Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4 = 0), 53, Springer (1968) | DOI

[3] C L Curtis, M H Freedman, W C Hsiang, R Stong, A decomposition theorem for h–cobordant smooth simply-connected compact 4–manifolds, Invent. Math. 123 (1996) 343 | DOI

[4] S K Donaldson, Irrationality and the h–cobordism conjecture, J. Differential Geom. 26 (1987) 141 | DOI

[5] S K Donaldson, The Seiberg–Witten equations and 4–manifold topology, Bull. Amer. Math. Soc. 33 (1996) 45 | DOI

[6] R Fintushel, R J Stern, Immersed spheres in 4–manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995) 145

[7] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 | DOI

[8] D Y Gan, J H Guo, Embeddings and immersions of a 2–sphere in 4–manifolds, Proc. Amer. Math. Soc. 118 (1993) 1323 | DOI

[9] H Z Gao, Representing homology classes of almost definite 4–manifolds, Topology Appl. 52 (1993) 109 | DOI

[10] H Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie (1952)

[11] M Kreck, h–cobordisms between 1–connected 4–manifolds, Geom. Topol. 5 (2001) 1 | DOI

[12] P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797 | DOI

[13] P B Kronheimer, T S Mrowka, Recurrence relations and asymptotics for four-manifold invariants, Bull. Amer. Math. Soc. 30 (1994) 215 | DOI

[14] K Kuga, Representing homology classes of S2 × S2, Topology 23 (1984) 133 | DOI

[15] R K Lashof, S Smale, Self-intersections of immersed manifolds, J. Math. Mech. 8 (1959) 143 | DOI

[16] T Lawson, Representing homology classes of almost definite 4–manifolds, Michigan Math. J. 34 (1987) 85 | DOI

[17] T Lawson, h–cobordisms between simply connected 4–manifolds, Topology Appl. 28 (1988) 75 | DOI

[18] T Lawson, The minimal genus problem, Expo. Math. 15 (1997) 385

[19] B H Li, Representing nonnegative homology classes of CP2nCP2 by minimal genus smooth embeddings, Trans. Amer. Math. Soc. 352 (2000) 4155 | DOI

[20] B H Li, T J Li, Minimal genus smooth embeddings in S2 × S2 and CP2nCP2 with n ≤ 8, Topology 37 (1998) 575 | DOI

[21] T J Li, A Liu, General wall crossing formula, Math. Res. Lett. 2 (1995) 797 | DOI

[22] R Matveyev, A decomposition of smooth simply-connected h–cobordant 4–manifolds, J. Differential Geom. 44 (1996) 571 | DOI

[23] J W Morgan, The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, 44, Princeton Univ. Press (1996)

[24] J W Morgan, Z Szabó, On h–cobordisms and Seiberg–Witten invariants, from: "Topics in symplectic –manifolds" (editor R J Stern), First Int. Press Lect. Ser. 1, International (1998) 117

[25] J W Morgan, Z Szabó, Complexity of 4–dimensional h–cobordisms, Invent. Math. 136 (1999) 273 | DOI

[26] J W Morgan, Z Szabó, C H Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706 | DOI

[27] R A Norman, Dehn’s lemma for certain 4–manifolds, Invent. Math. 7 (1969) 143 | DOI

[28] D Ruberman, The minimal genus of an embedded surface of non-negative square in a rational surface, Turkish J. Math. 20 (1996) 129

[29] S Smale, On the structure of manifolds, Amer. J. Math. 84 (1962) 387 | DOI

[30] J R Stallings, On infinite processes leading to differentiability in the complement of a point, from: "Differential and combinatorial topology" (editor S S Cairns), Princeton Univ. Press (1965) 245 | DOI

[31] S Strle, Bounds on genus and geometric intersections from cylindrical end moduli spaces, J. Differential Geom. 65 (2003) 469 | DOI

[32] A I Suciu, Immersed spheres in CP2 and S2 × S2, Math. Z. 196 (1987) 51 | DOI

[33] C T C Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962) 328 | DOI

[34] C T C Wall, Diffeomorphisms of 4–manifolds, J. Lond. Math. Soc. 39 (1964) 131 | DOI

[35] C T C Wall, On simply-connected 4–manifolds, J. Lond. Math. Soc. 39 (1964) 141 | DOI

[36] C T C Wall, On the orthogonal groups of unimodular quadratic forms, II, J. Reine Angew. Math. 213 (1964) 122 | DOI

[37] H Whitney, The self-intersections of a smooth n–manifold in 2n–space, Ann. of Math. 45 (1944) 220 | DOI

[38] X Zhao, H Gao, H Qiu, The minimal genus problem in rational surfaces CP2nCP2, Sci. China Ser. A 49 (2006) 1275 | DOI