## Geometry & Topology

### An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves

John Pardon

#### Abstract

We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from “counting” pseudo-holomorphic curves.

We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaf-theoretic tools play an important role in setting up our functorial algebraic “VFC package”.

We illustrate the methods we introduce by giving definitions of Gromov–Witten invariants and Hamiltonian Floer homology over $ℚ$ for general symplectic manifolds. Our framework generalizes to the $S1$–equivariant setting, and we use $S1$–localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, by Hofer and Salamon, by Ono, by Liu and Tian, by Ruan, and by Fukaya and Ono) is a well-known corollary of this calculation.

#### Article information

Source
Geom. Topol., Volume 20, Number 2 (2016), 779-1034.

Dates
Revised: 20 May 2015
Accepted: 1 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858971

Digital Object Identifier
doi:10.2140/gt.2016.20.779

Mathematical Reviews number (MathSciNet)
MR3493097

Zentralblatt MATH identifier
1342.53109

#### Citation

Pardon, John. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom. Topol. 20 (2016), no. 2, 779--1034. doi:10.2140/gt.2016.20.779. https://projecteuclid.org/euclid.gt/1510858971

#### References

• M Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres, Ann. of Math. 175 (2012) 71–185
• M Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent, Invent. Math. 189 (2012) 251–313
• J F Adams, On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 409–412
• F J Almgren, Jr, The homotopy groups of the integral cycle groups, Topology 1 (1962) 257–299
• M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA (1969)
• J-F Barraud, O Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. 166 (2007) 657–722
• N A Berikashvili, The Steenrod–Sitnikov homology theory on the category of compact spaces, Dokl. Akad. Nauk SSSR 254 (1980) 1289–1291
• N A Berikashvili, Axiomatics of the Steenrod–Sitnikov homology theory on the category of compact Hausdorff spaces, Trudy Mat. Inst. Steklov. 154 (1983) 24–37 In Russian; translated in Proc. Steklov Inst. Math. 154 (1984) 25–39
• R H Bing, The cartesian product of a certain nonmanifold and a line is $E\sp{4}$, Ann. of Math. 70 (1959) 399–412
• D Borisov, J Noel, Simplicial approach to derived differential manifolds, preprint (2011)
• J W Cannon, Shrinking cell-like decompositions of manifolds: codimension three, Ann. of Math. 110 (1979) 83–112
• G Carlsson, E K Pedersen, Čech homology and the Novikov conjectures for $K\!$– and $L$–theory, Math. Scand. 82 (1998) 5–47
• G S Chogoshvili, On the equivalence of the functional and spectral theory of homology, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951) 421–438
• R L Cohen, J D S Jones, G B Segal, Floer's infinite-dimensional Morse theory and homotopy theory, from: “The Floer memorial volume”, (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 297–325
• C C Conley, E Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V I Arnol'd, Invent. Math. 73 (1983) 33–49
• M Damian, On the stable Morse number of a closed manifold, Bull. London Math. Soc. 34 (2002) 420–430
• A Dress, Zur Spectralsequenz von Faserungen, Invent. Math. 3 (1967) 172–178
• D A Edwards, H M Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics 542, Springer, Berlin (1976)
• T Ekholm, I Smith, Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014) 195–240
• T Ekholm, I Smith, Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016) 1–59
• Y Eliashberg, Estimates on the number of fixed points of area preserving transformations, preprint, Syktyvar University (1979)
• A Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988) 393–407
• A Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775–813
• A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575–611
• A Floer, H Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993) 13–38
• A Floer, H Hofer, K Wysocki, Applications of symplectic homology, I, Math. Z. 217 (1994) 577–606
• E M Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies 104, Princeton Univ. Press (1982)
• K Fukaya, The symplectic $s$–cobordism conjecture: a summary, from: “Geometry and physics”, (J E Andersen, J Dupont, H Pedersen, A Swann, editors), Lecture Notes in Pure and Appl. Math. 184, Dekker, New York (1997) 209–219
• K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, I, AMS/IP Studies in Advanced Mathematics 46, Amer. Math. Soc., Providence, RI (2009)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, II, AMS/IP Studies in Advanced Mathematics 46, Amer. Math. Soc., Providence, RI (2009)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Technical details on Kuranishi structure and virtual fundamental chain, preprint (2012)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Kuranishi structure, pseudo-holomorphic curve, and virtual fundamental chain, Part 1, preprint (2015)
• K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933–1048
• R Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958)
• B Goldfarb, Novikov conjectures for arithmetic groups with large actions at infinity, $K$-Theory 11 (1997) 319–372
• M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• M Gromov, Singularities, expanders and topology of maps, Part $2$: From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal. 20 (2010) 416–526
• H M Hastings, Steenrod homotopy theory, homotopy idempotents, and homotopy limits, from: “Proceedings of the 1977 Topology Conference, II”, (W Kuperberg, G M Reed, P Zenor, editors), Topology Proc. 2 (1977) 461–477
• A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
• H Hofer, D A Salamon, Floer homology and Novikov rings, from: “The Floer memorial volume”, (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 483–524
• H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, I: A splicing-based differential geometry, J. Eur. Math. Soc. $($JEMS$)$ 9 (2007) 841–876
• H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, II: Implicit function theorems, Geom. Funct. Anal. 19 (2009) 206–293
• H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, III: Fredholm functors and polyfolds, Geom. Topol. 13 (2009) 2279–2387
• H Hofer, K Wysocki, E Zehnder, Integration theory on the zero sets of polyfold Fredholm sections, Math. Ann. 346 (2010) 139–198
• H Hofer, K Wysocki, E Zehnder, sc-smoothness, retractions and new models for smooth spaces, Discrete Contin. Dyn. Syst. 28 (2010) 665–788
• H Hofer, K Wysocki, E Zehnder, Applications of polyfold theory, I: The polyfolds of Gromov–Witten theory, preprint (2014)
• H Hofer, K Wysocki, E Zehnder, Polyfold and Fredholm theory, I: Basic theory in $M$–polyfolds, preprint (2014)
• H Inassaridze, On the Steenrod homology theory of compact spaces, Michigan Math. J. 38 (1991) 323–338
• D Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry, Book manuscript (2012) Available at \setbox0\makeatletter\@url https://people.maths.ox.ac.uk/joyce/dmbook.pdf {\unhbox0
• D Joyce, An introduction to d-manifolds and derived differential geometry, from: “Moduli spaces”, (Cambridge, editor), London Math. Soc. Lecture Note Ser. 411, Cambridge Univ. Press (2014) 230–281
• M Kontsevich, Enumeration of rational curves via torus actions, from: “The moduli space of curves”, (R Dijkgraaf, C Faber, G van der Geer, editors), Progr. Math. 129, Birkhäuser, Boston (1995) 335–368
• M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562
• J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: “Topics in symplectic $4$–manifolds”, (R J Stern, editor), First Int. Press Lect. Ser. 1, Int. Press, Cambridge, MA (1998) 47–83
• G Liu, G Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998) 1–74
• R B Lockhart, R C McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985) 409–447
• G Lu, G Tian, Constructing virtual Euler cycles and classes, Int. Math. Res. Surv. 2007 (2007)
• J Lurie, Higher topos theory (2012) Updated online version of Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press, 2009 Available at \setbox0\makeatletter\@url http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf {\unhbox0
• J Lurie, Higher algebra (2014) Available at \setbox0\makeatletter\@url http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf {\unhbox0
• S Mardešić, Strong shape and homology, Springer, Berlin (2000)
• D McDuff, Notes on Kuranishi atlases, preprint (2015)
• D McDuff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, Amer. Math. Soc., Providence, RI (1994)
• D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, RI (2004)
• D McDuff, K Wehrheim, Kuranishi atlases with trivial isotropy, preprint (2015)
• J Milnor, On the Steenrod homology theory, from: “Novikov conjectures, index theorems and rigidity, Vol. 1”, (S C Ferry, A Ranicki, J Rosenberg, editors), London Math. Soc. Lecture Note Ser. 226, Cambridge Univ. Press (1995) 79–96
• M-P Muller, Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves, from: “Holomorphic curves in symplectic geometry”, (M Audin, J Lafontaine, editors), Progr. Math. 117, Birkhäuser, Basel (1994) 217–231
• K Ono, On the Arnol'd conjecture for weakly monotone symplectic manifolds, Invent. Math. 119 (1995) 519–537
• M S Osborne, Basic homological algebra, Graduate Texts in Mathematics 196, Springer, New York (2000)
• J Pardon, The Hilbert–Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013) 879–899
• S Piunikhin, D Salamon, M Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, from: “Contact and symplectic geometry”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171–200
• J J Rotman, An introduction to homological algebra, 2nd edition, Springer, New York (2009)
• Y Ruan, Virtual neighborhoods and pseudo-holomorphic curves, from: “Proceedings of 6th Gökova Geometry-Topology Conference”, Turkish J. Math. 23 (1999) 161–231
• D Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology”, (Y Eliashberg, L Traynor, editors), IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, RI (1999) 143–229
• D Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303–1360
• N Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988) 121–154
• D I Spivak, Derived smooth manifolds, Duke Math. J. 153 (2010) 55–128
• N E Steenrod, Regular cycles of compact metric spaces, Ann. of Math. 41 (1940) 833–851
• M G Sullivan, $K\!$–theoretic invariants for Floer homology, Geom. Funct. Anal. 12 (2002) 810–872
• W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
• K Wehrheim, Energy quantization and mean value inequalities for nonlinear boundary value problems, J. Eur. Math. Soc. 7 (2005) 305–318
• K Wehrheim, Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing, from: “Proceedings of the Freedman Fest”, (R Kirby, V Krushkal, Z Wang, editors), Geom. Topol. Monogr. 18 (2012) 369–450