AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Cornered Heegaard Floer Homology
About this Title
Christopher L. Douglas, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK, Robert Lipshitz, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA and Ciprian Manolescu, Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 262, Number 1266
ISBNs: 978-1-4704-3771-8 (print); 978-1-4704-5505-7 (online)
DOI: https://doi.org/10.1090/memo/1266
Published electronically: December 23, 2019
MSC: Primary 57R56, 57R58
Table of Contents
Chapters
- 1. Introduction
- 2. Some abstract 2-algebra
- 3. More 2-algebra: Bending and smoothing
- 4. Some homological algebra of 2-modules
- 5. The algebras and algebra-modules
- 6. The cornering module–2-modules
- 7. The trimodules $\mathsf {T}_{DDD}$ and $\mathsf {T}_{DDA}$
- 8. Cornered 2-modules for cornered Heegaard diagrams
- 9. Gradings
- 10. Practical computations
- 11. The nilCoxeter planar algebra
Abstract
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.- Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527
- Christopher L. Douglas and Ciprian Manolescu, On the algebra of cornered Floer homology, J. Topol. 7 (2014), no. 1, 1–68. MR 3180613, DOI 10.1112/jtopol/jtt013
- Charles Ehresmann, Catégories doubles et catégories structurées, C. R. Acad. Sci. Paris 256 (1963), 1198–1201 (French). MR 152561
- Klaus Jänich, On the classification of $O(n)$-manifolds, Math. Ann. 176 (1968), 53–76. MR 226674, DOI 10.1007/BF02052956
- V. F. R. Jones, Planar algebras, I, 1999, arXiv:math/9909027.
- Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
- R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, Bordered Heegaard Floer homology: Invariance and pairing, 2008, arXiv:0810.0687.
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Heegaard Floer homology as morphism spaces, Quantum Topol. 2 (2011), no. 4, 381–449. MR 2844535, DOI 10.4171/qt/25
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Computing $\widehat {HF}$ by factoring mapping classes, Geom. Topol. 18 (2014), no. 5, 2547–2681. MR 3285222, DOI 10.2140/gt.2014.18.2547
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bimodules in bordered Heegaard Floer homology, Geom. Topol. 19 (2015), no. 2, 525–724. MR 3336273, DOI 10.2140/gt.2015.19.525
- J. Lurie, Derived algebraic geometry VI: $E_k$ algebras, 2009, \eprint{arXiv:0911.0018}.
- Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
- Scott Morrison and Kevin Walker, Blob homology, Geom. Topol. 16 (2012), no. 3, 1481–1607. MR 2978449, DOI 10.2140/gt.2012.16.1481
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
- Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356, DOI 10.1016/j.aim.2005.03.014
- N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
- R. Zarev, Bordered Floer homology for sutured manifolds, 2009, \eprint{arXiv:0908.1106}.
- R. Zarev, Joining and gluing sutured Floer homology, 2010, \eprint{arXiv:1010.3496}.