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In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Hiro Lee Tanaka
From the journal Advances in Geometry


Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T*Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

MSC 2010: 53D12; 53D37

Funding statement: The author was supported by a Presidential Fellowship from Northwestern University’s Office of the President, an NSF Graduate Research Fellowship, and a Mathematical Sciences Research Institute Postdoctoral Fellowship.


We are grateful to Tim Perutz for helpful feedback on this paper. We also thank the referee for very helpful comments.

  1. Communicated by: K. Ono


[1] M. Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189 (2012), 251–313. MR2947545 Zbl 1261.53077Search in Google Scholar

[2] V. I. Arnol’d, The first steps of symplectic topology (Russian). Uspekhi Mat. Nauk41 (1986), no. 6 (252), 3–18, 229. English translation: Russian Math. Surveys41 (1986), no. 6, 1–21. MR890489 Zbl 0618.58021Search in Google Scholar

[3] P. Biran, O. Cornea, Lagrangian cobordism. I. J. Amer. Math. Soc. 26 (2013), 295–340. MR3011416 Zbl 1272.53071Search in Google Scholar

[4] P. Biran, O. Cornea, Lagrangian cobordism. II. Preprint 2013, arXiv:1304.6032 [math.SG]Search in Google Scholar

[5] K. Fukaya, P. Seidel, I. Smith, Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172 (2008), 1–27. MR2385665 Zbl 1140.53036Search in Google Scholar

[6] T. Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture. Geom. Topol. 17 (2013), 639–731. MR3070514 Zbl 1267.53081Search in Google Scholar

[7] J. Milnor, Lectures on the h-cobordism theorem. Princeton Univ. Press 1965. MR0190942 Zbl 0161.20302Search in Google Scholar

[8] D. Nadler, Microlocal branes are constructible sheaves. Selecta Math. \U(N.S.\U)15 (2009), 563–619. MR2565051 Zbl 1197.53116Search in Google Scholar

[9] D. Nadler, H. L. Tanaka, A stable ∞-category of Lagrangian cobordisms. Preprint 2011, arXiv:1109.4835 [math.SG]Search in Google Scholar

[10] S. Smale, On the structure of manifolds. Amer. J. Math. 84 (1962), 387–399. MR0153022 Zbl 0109.41103Search in Google Scholar

[11] L. S. Suárez, Exact Lagrangian cobordism and pseudo-isotopy. Internat. J. Math. 28 (2017), 1750059, 35pp. MR3681121 Zbl 1379.53095Search in Google Scholar

[12] H. L. Tanaka, The Fukaya category pairs with Lagrangian cobordisms. Preprint 2016, arXiv:1607.04976 [math.SG]Search in Google Scholar

Received: 2018-09-07
Revised: 2018-12-08
Published Online: 2019-09-11
Published in Print: 2021-01-27

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