Unfolded Seiberg–Witten Floer spectra, I : Definition and invariance

Khandhawit, Tirasan¹ ; Lin, Jianfeng² ; Sasahira, Hirofumi³

¹ Kavli IPMU, The University of Tokyo, Kashiwa, Japan
² Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
³ Faculty of Mathematics, Kyushu University, Fukuoka, Japan

Geometry & topology, Tome 22 (2018) no. 4, pp. 2027-2114.

Résumé

Let $Y$ be a closed and oriented $3$–manifold. We define different versions of unfolded Seiberg–Witten Floer spectra for $Y$. These invariants generalize Manolescu’s Seiberg–Witten Floer spectrum for rational homology $3$–spheres. We also compute some examples when $Y$ is a Seifert space.

Détail

DOI : 10.2140/gt.2018.22.2027

Classification : 57R57, 57R58
Keywords: 3-manifolds, Floer homotopy, Seiberg–Witten theory, Conley index

Bibliographie

[1] J F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, from: "Algebraic topology" (editors I Madsen, B Oliver), Lecture Notes in Math. 1051, Springer (1984) 483 | DOI

[2] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 | DOI

[3] S Bauer, A stable cohomotopy refinement of Seiberg–Witten invariants, II, Invent. Math. 155 (2004) 21 | DOI

[4] 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 | DOI

[5] C Conley, Isolated invariant sets and the Morse index, 38, Amer. Math. Soc. (1978)

[6] O Cornea, Homotopical dynamics: suspension and duality, Ergodic Theory Dynam. Systems 20 (2000) 379 | DOI

[7] T Tom Dieck, Transformation groups, 8, de Gruyter (1987) | DOI

[8] A Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems 7 (1987) 93 | DOI

[9] A Floer, An instanton-invariant for 3–manifolds, Comm. Math. Phys. 118 (1988) 215

[10] M Furuta, Monopole equation and the –conjecture, Math. Res. Lett. 8 (2001) 279 | DOI

[11] M Furuta, T J Li, Intersection form of spin 4–manifolds with boundary, preprint (2014)

[12] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[13] T Khandhawit, Twisted Manolescu–Floer spectra for Seiberg–Witten monopoles, PhD thesis, MIT (2013)

[14] T Khandhawit, J Lin, H Sasahira, The unfolded Seiberg–Witten–Floer spectra, II, in preparation

[15] T Khandhawit, J Lin, H Sasahira, The unfolded Seiberg–Witten–Floer spectra, III, in preparation

[16] P B Kronheimer, C Manolescu, Periodic Floer pro-spectra from the Seiberg–Witten equations, preprint (2002)

[17] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007) | DOI

[18] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, 1213, Springer (1986) | DOI

[19] T Lidman, C Manolescu, The equivalence of two Seiberg–Witten Floer homologies, preprint (2016)

[20] F Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture, preprint (2014)

[21] J Lin, Pin(2)–equivariant KO-theory and intersection forms of spin 4–manifolds, Algebr. Geom. Topol. 15 (2015) 863 | DOI

[22] C Manolescu, Seiberg–Witten–Floer stable homotopy type of three-manifolds with b1 = 0, Geom. Topol. 7 (2003) 889 | DOI

[23] C Manolescu, On the intersection forms of spin four-manifolds with boundary, Math. Ann. 359 (2014) 695 | DOI

[24] C Manolescu, Pin(2)–equivariant Seiberg–Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016) 147 | DOI

[25] J P May, Equivariant homotopy and cohomology theory, 91, Amer. Math. Soc. (1996) | DOI

[26] T Mrowka, P Ozsváth, B Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997) 685 | DOI

[27] L I Nicolaescu, Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg–Witten moduli spaces, Israel J. Math. 114 (1999) 61 | DOI

[28] L I Nicolaescu, Finite energy Seiberg–Witten moduli spaces on 4–manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8 (2000) 1027 | DOI

[29] M Q Ouyang, Geometric invariants for Seifert fibred 3–manifolds, Trans. Amer. Math. Soc. 346 (1994) 641 | DOI

[30] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159 | DOI

[31] A M Pruszko, The Conley index for flows preserving generalized symmetries, from: "Conley index theory" (editors K Mischaikow, M Mrozek, P Zgliczyński), Banach Center Publ. 47, Polish Acad. Sci. Inst. Math. (1999) 193

[32] K P Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc. 269 (1982) 351 | DOI

[33] D Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985) 1 | DOI

[34] H Sasahira, Gluing formula for the stable cohomotopy version of Seiberg–Witten invariants along 3–manifolds with b1 > 0, preprint (2014)

[35] E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769 | DOI