Geometry & Topology Volume 29, issue 3 (2025)

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Exotic Dehn twists on sums of two contact $3$-manifolds

Eduardo Fernández and Juan Muñoz-Echániz

Geometry & Topology 29 (2025) 1571–1618

DOI: 10.2140/gt.2025.29.1571

Bibliography

1	R H Bamler, B Kleiner, Ricci flow and contractibility of spaces of metrics, preprint (2019) arXiv:1909.08710
2	R H Bamler, B Kleiner, Ricci flow and diffeomorphism groups of 3-manifolds, J. Amer. Math. Soc. 36 (2023) 563 MR4536904
3	R H Bamler, B Kleiner, Diffeomorphism groups of prime 3-manifolds, J. Reine Angew. Math. 806 (2024) 23 MR4685084
4	J Bodnár, O Plamenevskaya, Heegaard Floer invariants of contact structures on links of surface singularities, Quantum Topol. 12 (2021) 411 MR4321210
5	M S Borman, Y Eliashberg, E Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015) 281 MR3455235
6	F Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett. 13 (2006) 71 MR2200047
7	R Casals, D M Pancholi, F Presas, The Legendrian Whitney trick, Geom. Topol. 25 (2021) 3229 MR4347315
8	J Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ₄ = 0), 53, Springer (1968) MR229250
9	V Colin, Chirurgies d’indice un et isotopies de sphères dans les variétés de contact tendues, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 659 MR1447038
10	V Colin, P Ghiggini, K Honda, Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011) 8100 MR2806645
11	J Conway, H Min, Classification of tight contact structures on surgeries on the figure-eight knot, Geom. Topol. 24 (2020) 1457 MR4157557
12	F Ding, H Geiges, A unique decomposition theorem for tight contact 3-manifolds, Enseign. Math. 53 (2007) 333 MR2455948
13	F Ding, H Geiges, The diffeotopy group of S¹ × S² via contact topology, Compos. Math. 146 (2010) 1096 MR2660686
14	M Echeverria, Naturality of the contact invariant in monopole Floer homology under strong symplectic cobordisms, Algebr. Geom. Topol. 20 (2020) 1795 MR4127085
15	Y Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989) 623 MR1022310
16	Y Eliashberg, Contact 3-manifolds twenty years since J Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165 MR1162559
17	Y Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, from: "Topological methods in modern mathematics" (editors L R Goldberg, A V Phillips), Publish or Perish (1993) 171 MR1215964
18	Y Eliashberg, N Mishachev, The space of tight contact structures on ℝ³ is contractible, preprint (2021) arXiv:2108.09452
19	E Fernández, J Martínez-Aguinaga, F Presas, The homotopy type of the contactomorphism groups of tight contact 3-manifolds, I, preprint (2020) arXiv:2012.14948
20	D Fuchs, S Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997) 1025 MR1445553
21	D Gabai, The Smale conjecture for hyperbolic 3-manifolds : Isom(M³) ≃ Diff(M³), J. Differential Geom. 58 (2001) 113 MR1895350
22	H Geiges, An introduction to contact topology, 109, Cambridge Univ. Press (2008) MR2397738
23	H Geiges, J Gonzalo Perez, On the topology of the space of contact structures on torus bundles, Bull. London Math. Soc. 36 (2004) 640 MR2070440
24	H Geiges, M Klukas, The fundamental group of the space of contact structures on the 3-torus, Math. Res. Lett. 21 (2014) 1257 MR3335846
25	F Gironella, Examples of contact mapping classes of infinite order in all dimensions, Math. Res. Lett. 28 (2021) 707 MR4270271
26	E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637 MR1129802
27	E Giroux, Une infinité de structures de contact tendues sur une infinité de variétés, Invent. Math. 135 (1999) 789 MR1669264
28	E Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001) 218 MR1839346
29	E Giroux, P Massot, On the contact mapping class group of Legendrian circle bundles, Compos. Math. 153 (2017) 294 MR3604864
30	R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619 MR1668563
31	M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 MR809718
32	M Gromov, Partial differential relations, 9, Springer (1986) MR864505
33	V L Hansen, The homotopy type of the space of maps of a homology 3-sphere into the 2-sphere, Pacific J. Math. 76 (1978) 43 MR499632
34	A Hatcher, Homeomorphisms of sufficiently large P²-irreducible 3-manifolds, Topology 15 (1976) 343 MR420620
35	A Hatcher, On the diffeomorphism group of S¹ × S², Proc. Amer. Math. Soc. 83 (1981) 427 MR624946
36	A E Hatcher, A proof of the Smale conjecture, Diff(S³) ≃ O(4), Ann. of Math. 117 (1983) 553 MR701256
37	A Hatcher, N Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J. 155 (2010) 205 MR2736166
38	K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002) 435 MR1940409
39	S Hong, J Kalliongis, D McCullough, J H Rubinstein, Diffeomorphisms of elliptic 3-manifolds, 2055, Springer (2012) MR2976322
40	N V Ivanov, Groups of diffeomorphisms of Waldhausen manifolds, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 66 (1976) 172 MR448370
41	Y Kanda, The classification of tight contact structures on the 3-torus, Comm. Anal. Geom. 5 (1997) 413 MR1487723
42	P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209 MR1474156
43	P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007) MR2388043
44	P B Kronheimer, T S Mrowka, The Dehn twist on a sum of two K3 surfaces, Math. Res. Lett. 27 (2020) 1767 MR4216604
45	P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007) 457 MR2299739
46	Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology, Geom. Topol. 24 (2020) 2829 MR4194305
47	F Lin, Lectures on monopole Floer homology, from: "Proceedings of the Gökova geometry–topology conference 2015" (editors S Akbulut, D Auroux, T Önder), GGT (2016) 39 MR3526838
48	J Lin, Isotopy of the Dehn twist on K3 # K3 after a single stabilization, Geom. Topol. 27 (2023) 1987 MR4621924
49	D McCullough, Homotopy groups of the space of self-homotopy-equivalences, Trans. Amer. Math. Soc. 264 (1981) 151 MR597873
50	D McCullough, Topological and algebraic automorphisms of 3-manifolds, from: "Groups of self-equivalences and related topics" (editor R A Piccinini), Lecture Notes in Math. 1425, Springer (1990) 102 MR1070580
51	D McCullough, T Soma, The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold, J. Differential Geom. 93 (2013) 327 MR3024309
52	J Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959) 272 MR100267
53	H Min, The contact mapping class group and rational unknots in lens spaces, Int. Math. Res. Not. 2024 (2024) 11315 MR4782801
54	T Mrowka, Y Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006) 1 MR2199446
55	J Muñoz-Echániz, A monopole invariant for families of contact structures, Adv. Math. 439 (2024) 109483 MR4687910
56	A Némethi, Links of rational singularities, L-spaces and LO fundamental groups, Invent. Math. 210 (2017) 69 MR3698339
57	W D Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299 MR632532
58	P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39 MR2153455
59	R S Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966) 1 MR189028
60	P Seidel, Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999) 145 MR1743463
61	P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103 MR1765826
62	P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003 MR1978046
63	G Smirnov, Seidel’s theorem via gauge theory, preprint (2020) arXiv:2010.03361
64	G Smirnov, Symplectic mapping class groups of K3 surfaces and Seiberg–Witten invariants, Geom. Funct. Anal. 32 (2022) 280 MR4408433
65	C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809 MR1306023
66	C H Taubes, SW ⇒ Gr : from the Seiberg–Witten equations to pseudo-holomorphic curves, from: "Seiberg Witten and Gromov invariants for symplectic 4-manifolds" (editor R Wentworth), First Int. Press Lect. Ser. 2, International (2000) 1 MR1798137
67	T Vogel, Non-loose unknots, overtwisted discs, and the contact mapping class group of S³, Geom. Funct. Anal. 28 (2018) 228 MR3777417
68	T Watanabe, Some exotic nontrivial elements of the rational homotopy groups of Diff(S⁴), preprint (2018) arXiv:1812.02448
69	M Weiss, What does the classifying space of a category classify?, Homology Homotopy Appl. 7 (2005) 185 MR2175298