Algebraic & Geometric Topology Volume 24, issue 3 (2024)

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Morse–Bott cohomology from homological perturbation theory

Zhengyi Zhou

Algebraic & Geometric Topology 24 (2024) 1321–1429

DOI: 10.2140/agt.2024.24.1321

Bibliography

1	M Abouzaid, Symplectic cohomology and Viterbo’s theorem, from: "Free loop spaces in geometry and topology" (editors J Latschev, A Oancea), IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc. (2015) 271 MR3444367
2	M Audin, M Damian, Morse theory and Floer homology, Springer (2014) MR3155456
3	D M Austin, P J Braam, Morse–Bott theory and equivariant cohomology, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 123 MR1362827
4	A Banyaga, D E Hurtubise, Morse–Bott homology, Trans. Amer. Math. Soc. 362 (2010) 3997 MR2608393
5	A Banyaga, D E Hurtubise, Cascades and perturbed Morse–Bott functions, Algebr. Geom. Topol. 13 (2013) 237 MR3031642
6	J F Barraud, O Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. 166 (2007) 657 MR2373371
7	P Biran, O Cornea, A Lagrangian quantum homology, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 1 MR2555932
8	R Bott, Nondegenerate critical manifolds, Ann. of Math. 60 (1954) 248 MR0064399
9	R Bott, An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84 (1956) 251 MR0087035
10	R Bott, H Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964 MR0105694
11	R Bott, L W Tu, Differential forms in algebraic topology, 82, Springer (1982) MR0658304
12	F Bourgeois, A Morse–Bott approach to contact homology, PhD thesis, Stanford University (2002) MR2703292
13	F Bourgeois, K Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004) 123 MR2092725
14	F Bourgeois, A Oancea, An exact sequence for contact- and symplectic homology, Invent. Math. 175 (2009) 611 MR2471597
15	F Bourgeois, A Oancea, Symplectic homology, autonomous Hamiltonians, and Morse–Bott moduli spaces, Duke Math. J. 146 (2009) 71 MR2475400
16	F Bourgeois, A Oancea, The Gysin exact sequence for S¹–equivariant symplectic homology, J. Topol. Anal. 5 (2013) 361 MR3152208
17	K Cieliebak, K Mohnke, Symplectic hypersurfaces and transversality in Gromov–Witten theory, J. Symplectic Geom. 5 (2007) 281 MR2399678
18	K Cieliebak, E Volkov, Chern–Simons theory and string topology, preprint (2023) arXiv:2312.05922
19	R L Cohen, J D S Jones, G B Segal, Floer’s infinite-dimensional Morse theory and homotopy theory, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 297 MR1362832
20	L Diogo, S T Lisi, Morse–Bott split symplectic homology, J. Fixed Point Theory Appl. 21 (2019) 77 MR3981446
21	L Diogo, S T Lisi, Symplectic homology of complements of smooth divisors, J. Topol. 12 (2019) 967 MR4072162
22	S Dostoglou, D A Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. 139 (1994) 581 MR1283871
23	Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: "Visions in mathematics" (editors N Alon, J Bourgain, A Connes, M Gromov, V Milman), Birkhäuser (= GAFA special volume) (2000) 560 MR1826267
24	O Fabert, J W Fish, R Golovko, K Wehrheim, Polyfolds: a first and second look, EMS Surv. Math. Sci. 3 (2016) 131 MR3576532
25	B Filippenko, Polyfold regularization of constrained moduli spaces, J. Symplectic Geom. 19 (2021) 241 MR4325406
26	J Fish, H Hofer, Applications of polyfold theory, II: The polyfolds of symplectic field theory, in preparation
27	A Floer, An instanton-invariant for 3–manifolds, Comm. Math. Phys. 118 (1988) 215 MR0956166
28	A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513 MR0965228
29	A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575 MR0987770
30	A Floer, Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989) 207 MR1001276
31	A Floer, H Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993) 13 MR1200162
32	U Frauenfelder, The Arnold–Givental conjecture and moment Floer homology, Int. Math. Res. Not. 2004 (2004) 2179 MR2076142
33	K Fukaya, Floer homology of connected sum of homology 3–spheres, Topology 35 (1996) 89 MR1367277
34	K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, II, 46.2, Amer. Math. Soc. (2009) MR2548482
35	K Fukaya, Y G Oh, H Ohta, K Ono, Kuranishi structure, pseudo-holomorphic curve, and virtual fundamental chain, I, preprint (2015) arXiv:1503.07631
36	P Griffiths, J Harris, Principles of algebraic geometry, Wiley (1978) MR507725
37	M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 MR0809718
38	V W Guillemin, S Sternberg, Supersymmetry and equivariant de Rham theory, Springer (1999) MR1689252
39	F R Harvey, H B Lawson Jr., Finite volume flows and Morse theory, Ann. of Math. 153 (2001) 1 MR1826410
40	H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, I : A splicing-based differential geometry, J. Eur. Math. Soc. 9 (2007) 841 MR2341834
41	H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, II : Implicit function theorems, Geom. Funct. Anal. 19 (2009) 206 MR2507223
42	H Hofer, K Wysocki, E Zehnder, A general Fredholm theory, III : Fredholm functors and polyfolds, Geom. Topol. 13 (2009) 2279 MR2515707
43	H Hofer, K Wysocki, E Zehnder, Integration theory on the zero sets of polyfold Fredholm sections, Math. Ann. 346 (2010) 139 MR2558891
44	H Hofer, K Wysocki, E Zehnder, Polyfold and Fredholm theory, 72, Springer (2021) MR4298268
45	D E Hurtubise, Multicomplexes and spectral sequences, J. Algebra Appl. 9 (2010) 519 MR2718643
46	E N Ionel, T H Parker, A natural Gromov–Witten virtual fundamental class, preprint (2013) arXiv:1302.3472
47	D Joyce, On manifolds with corners, from: "Advances in geometric analysis" (editors S Janeczko, J Li, D H Phong), Adv. Lect. Math. 21, International (2012) 225 MR3077259
48	D Joyce, A new definition of Kuranishi space, preprint (2014) arXiv:1409.6908
49	M Kontsevich, Y Soibelman, Homological mirror symmetry and torus fibrations, from: "Symplectic geometry and mirror symmetry" (editors K Fukaya, Y G Oh, K Ono, G Tian), World Sci. (2001) 203 MR1882331
50	P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007) MR2388043
51	J Latschev, Gradient flows of Morse–Bott functions, Math. Ann. 318 (2000) 731 MR1802508
52	J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: "Topics in symplectic 4–manifolds" (editor R J Stern), First Int. Press Lect. Ser. 1, International (1998) 47 MR1635695
53	F Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture, 1221, Amer. Math. Soc. (2018) MR3827053
54	R Lipshitz, S Sarkar, A Khovanov stable homotopy type, J. Amer. Math. Soc. 27 (2014) 983 MR3230817
55	J McCleary, User’s guide to spectral sequences, 12, Publish or Perish (1985) MR820463
56	D McDuff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004) MR2045629
57	D McDuff, K Wehrheim, Smooth Kuranishi atlases with isotropy, Geom. Topol. 21 (2017) 2725 MR3687107
58	R Melrose, Differential analysis on manifolds with corners, unpublished manuscript (1996)
59	M Morse, The calculus of variations in the large, 18, Amer. Math. Soc. (1932)
60	D Murfet, Derived categories, I, preprint (2006)
61	L I Nicolaescu, An invitation to Morse theory, Springer (2007) MR2298610
62	P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027 MR2113019
63	J Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol. 20 (2016) 779 MR3493097
64	J Pardon, Contact homology and virtual fundamental cycles, J. Amer. Math. Soc. 32 (2019) 825 MR3981989
65	L Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, J. Topol. Anal. 2 (2010) 469 MR2748215
66	J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827 MR1241874
67	Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259 MR1366548
68	F Schmäschke, Floer homology of Lagrangians in clean intersection, preprint (2016) arXiv:1606.05327
69	M Schwarz, Morse homology, 111, Birkhäuser (1993) MR1239174
70	P Seidel, A biased view of symplectic cohomology, from: "Current developments in mathematics" (editors B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2008) 211 MR2459307
71	P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc. (2008) MR2441780
72	E G Sklyarenko, The Thom isomorphism for nonorientable bundles, Fundam. Prikl. Mat. 9 (2003) 55 MR2093413
73	K Wehrheim, Fredholm notions in scale calculus and Hamiltonian Floer theory, (2012) arXiv:1209.4040
74	K Wehrheim, Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing, from: "Proceedings of the Freedman Fest" (editors R Kirby, V Krushkal, Z Wang), Geom. Topol. Monogr. 18, Geom. Topol. Publ. (2012) 369 MR3084244
75	C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) MR1269324
76	E Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982) 661 MR0683171
77	Z Zhou, Equivariant fundamental class and localization theorem, in preparation
78	Z Zhou, Quotient theorems in polyfold theory and S¹–equivariant transversality, Proc. Lond. Math. Soc. 121 (2020) 1337 MR4133710
79	Z Zhou, On the cohomology ring of symplectic fillings, Algebr. Geom. Topol. 23 (2023) 1693 MR4602411
80	A Zinger, The determinant line bundle for Fredholm operators : construction, properties, and classification, Math. Scand. 118 (2016) 203 MR3515189