AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Naturality and Mapping Class Groups in Heegaard Floer Homology
About this Title
András Juhász, Dylan P. Thurston and Ian Zemke
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 273, Number 1338
ISBNs: 978-1-4704-4972-8 (print); 978-1-4704-6805-7 (online)
DOI: https://doi.org/10.1090/memo/1338
Published electronically: November 8, 2021
Keywords: Heegaard Floer homology,
3-manifold,
Heegaard diagram
Table of Contents
Chapters
- 1. Introduction
- 2. Heegaard Invariants
- 3. Examples
- 4. Singularities of Smooth Functions
- 5. Generic One- and Two-Parameter Families of Gradients
- 6. Translating Bifurcations of Gradients to Heegaard Diagrams
- 7. Simplifying Moves on Heegaard Diagrams
- 8. Simplifying Handleswaps
- 9. Strong Heegaard Invariants Have No Monodromy
- 10. Heegaard Floer Homology
- A. The 2-complex of Handleslides
Abstract
We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant.
The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.
- Casim Abbas, An introduction to compactness results in symplectic field theory, Springer, Heidelberg, 2014. MR 3157146, DOI 10.1007/978-3-642-31543-5
- D. V. Anosov, S. Kh. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines, and Yu. S. Il′yashenko, Ordinary differential equations and smooth dynamical systems, Springer-Verlag, Berlin, 1997. Translated from the 1985 Russian original by E. R. Dawson and D. O’Shea; Third printing of the 1988 translation [Dynamical systems. I, Encyclopaedia Math. Sci., 1, Springer, Berlin, 1988; MR0970793 (89g:58060)]. MR 1633529
- V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil′ev, Singularity theory. I, Springer-Verlag, Berlin, 1998. Translated from the 1988 Russian original by A. Iacob; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. VI, Encyclopaedia Math. Sci., 6, Springer, Berlin, 1993; MR1230637 (94b:58018)]. MR 1660090, DOI 10.1007/978-3-642-58009-3
- Gregory Arone and Marja Kankaanrinta, On the functoriality of the blow-up construction, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 821–832. MR 2777772
- Francis Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237–270. MR 732345
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- M. J. Dias Carneiro and J. Palis, Bifurcations and global stability of families of gradients, Inst. Hautes Études Sci. Publ. Math. 70 (1989), 103–168 (1990). MR 1067381
- Jean Cerf, The pseudo-isotopy theorem for simply connected differentiable manifolds, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1970, pp. 76–82. MR 0290404
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
- Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
- David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
- J. Elisenda Grigsby and Stephan M. Wehrli, On the colored Jones polynomial, sutured Floer homology, and knot Floer homology, Adv. Math. 223 (2010), no. 6, 2114–2165. MR 2601010, DOI 10.1016/j.aim.2009.11.002
- Kristen Hendricks and Ciprian Manolescu, Involutive Heegaard Floer homology, Duke Math. J. 166 (2017), no. 7, 1211–1299. MR 3649355, DOI 10.1215/00127094-3793141
- Jesse Johnson and Darryl McCullough, The space of Heegaard splittings, J. Reine Angew. Math. 679 (2013), 155–179. MR 3065157, DOI 10.1515/crelle.2012.016
- András Juhász and Marco Marengon, Concordance maps in knot Floer homology, Geom. Topol. 20 (2016), no. 6, 3623–3673. MR 3590358, DOI 10.2140/gt.2016.20.3623
- András Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457. MR 2253454, DOI 10.2140/agt.2006.6.1429
- András Juhász, Cobordisms of sutured manifolds and the functoriality of link Floer homology, Adv. Math. 299 (2016), 940–1038. MR 3519484, DOI 10.1016/j.aim.2016.06.005
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
- Robert Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006), 955–1096. [Paging previously given as 955–1097]. MR 2240908, DOI 10.2140/gt.2006.10.955
- Robert Lipshitz, Correction to the article: A cylindrical reformulation of Heegaard Floer homology [MR2240908], Geom. Topol. 18 (2014), no. 1, 17–30. MR 3158771, DOI 10.2140/gt.2014.18.17
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree, J. Topol. 9 (2016), no. 2, 607–686. MR 3509974, DOI 10.1112/jtopol/jtw003
- Robert Lipshitz, Peter S. Ozsvath, and Dylan P. Thurston, Bordered Heegaard Floer homology, Mem. Amer. Math. Soc. 254 (2018), no. 1216, viii+279. MR 3827056, DOI 10.1090/memo/1216
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, 2nd ed., American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012. MR 2954391
- John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, DOI 10.4007/annals.2004.159.1159
- 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
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008), no. 2, 615–692. MR 2443092, DOI 10.2140/agt.2008.8.615
- Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 1–68. MR 2764036, DOI 10.2140/agt.2011.11.1
- J. Palis and F. Takens, Stability of parametrized families of gradient vector fields, Ann. of Math. (2) 118 (1983), no. 3, 383–421. MR 727698, DOI 10.2307/2006976
- Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683
- Sucharit Sarkar, Maslov index formulas for Whitney $n$-gons, J. Symplectic Geom. 9 (2011), no. 2, 251–270. MR 2811652
- Sucharit Sarkar, Moving basepoints and the induced automorphisms of link Floer homology, Algebr. Geom. Topol. 15 (2015), no. 5, 2479–2515. MR 3426686, DOI 10.2140/agt.2015.15.2479
- Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174, DOI 10.1007/978-3-0348-8577-5
- P. Seidel, $\pi _1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), no. 6, 1046–1095. MR 1487754, DOI 10.1007/s000390050037
- Gert Vegter, Global stability of generic two-parameter families of gradients on three-manifolds, Dynamical systems and bifurcations (Groningen, 1984) Lecture Notes in Math., vol. 1125, Springer, Berlin, 1985, pp. 107–129. MR 798085, DOI 10.1007/BFb0075638
- Bronisław Wajnryb, Mapping class group of a handlebody, Fund. Math. 158 (1998), no. 3, 195–228. MR 1663329, DOI 10.4064/fm-158-3-195-228
- Ian Michael Zemke, TQFT Constructions in Heegaard Floer Homology, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–University of California, Los Angeles. MR 3697764
- Ian Michael Zemke, TQFT Constructions in Heegaard Floer Homology, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–University of California, Los Angeles. MR 3697764
- Ian Zemke, Quasistabilization and basepoint moving maps in link Floer homology, Algebr. Geom. Topol. 17 (2017), no. 6, 3461–3518. MR 3709653, DOI 10.2140/agt.2017.17.3461
- Ian Zemke, Link cobordisms and functoriality in link Floer homology, J. Topol. 12 (2019), no. 1, 94–220. MR 3905679, DOI 10.1112/topo.12085