Geometry & Topology

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

Tirasan Khandhawit, Jianfeng Lin, and Hirofumi Sasahira

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Abstract

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.

Article information

Source
Geom. Topol., Volume 22, Number 4 (2018), 2027-2114.

Dates
Received: 20 May 2016
Revised: 18 June 2017
Accepted: 22 July 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584817

Digital Object Identifier
doi:10.2140/gt.2018.22.2027

Mathematical Reviews number (MathSciNet)
MR3784516

Zentralblatt MATH identifier
06864332

Subjects
Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 57R58: Floer homology

Keywords
3-manifolds Floer homotopy Seiberg–Witten theory Conley index

Citation

Khandhawit, Tirasan; Lin, Jianfeng; Sasahira, Hirofumi. Unfolded Seiberg–Witten Floer spectra, I: Definition and invariance. Geom. Topol. 22 (2018), no. 4, 2027--2114. doi:10.2140/gt.2018.22.2027. https://projecteuclid.org/euclid.gt/1523584817


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References

  • J F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's lecture, from “Algebraic topology” (I Madsen, B Oliver, editors), Lecture Notes in Math. 1051, Springer (1984) 483–532
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • S Bauer, A stable cohomotopy refinement of Seiberg–Witten invariants, II, Invent. Math. 155 (2004) 21–40
  • 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 Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence, RI (1978)
  • O Cornea, Homotopical dynamics: suspension and duality, Ergodic Theory Dynam. Systems 20 (2000) 379–391
  • T tom Dieck, Transformation groups, De Gruyter Studies in Mathematics 8, de Gruyter, Berlin (1987)
  • A Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems 7 (1987) 93–103
  • A Floer, An instanton-invariant for $3$–manifolds, Comm. Math. Phys. 118 (1988) 215–240
  • M Furuta, Monopole equation and the $\frac{11}8$–conjecture, Math. Res. Lett. 8 (2001) 279–291
  • M Furuta, T-J Li, Intersection form of spin $4$–manifolds with boundary, preprint (2014)
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • T Khandhawit, Twisted Manolescu–Floer spectra for Seiberg–Witten monopoles, PhD thesis, MIT (2013) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/1470984943 {\unhbox0
  • T Khandhawit, J Lin, H Sasahira, The unfolded Seiberg–Witten–Floer spectra, II, in preparation
  • T Khandhawit, J Lin, H Sasahira, The unfolded Seiberg–Witten–Floer spectra, III, in preparation
  • P B Kronheimer, C Manolescu, Periodic Floer pro-spectra from the Seiberg–Witten equations, preprint (2002)
  • P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press (2007)
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer (1986)
  • T Lidman, C Manolescu, The equivalence of two Seiberg–Witten Floer homologies, preprint (2016)
  • F Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture, preprint (2014) To appear in Mem. Amer. Math. Soc.
  • J Lin, $\mathrm{Pin}(2)$–equivariant KO-theory and intersection forms of spin $4$–manifolds, Algebr. Geom. Topol. 15 (2015) 863–902
  • C Manolescu, Seiberg–Witten–Floer stable homotopy type of three-manifolds with $b_1=0$, Geom. Topol. 7 (2003) 889–932
  • C Manolescu, On the intersection forms of spin four-manifolds with boundary, Math. Ann. 359 (2014) 695–728
  • C Manolescu, $\mathrm{Pin}(2)$–equivariant Seiberg–Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016) 147–176
  • J P May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics 91, Amer. Math. Soc., Providence, RI (1996)
  • T Mrowka, P Ozsváth, B Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997) 685–791
  • 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) 61–123
  • L I Nicolaescu, Finite energy Seiberg–Witten moduli spaces on $4$–manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8 (2000) 1027–1096
  • M Q Ouyang, Geometric invariants for Seifert fibred $3$–manifolds, Trans. Amer. Math. Soc. 346 (1994) 641–659
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • A M Pruszko, The Conley index for flows preserving generalized symmetries, from “Conley index theory” (K Mischaikow, M Mrozek, P Zgliczyński, editors), Banach Center Publ. 47, Polish Acad. Sci. Inst. Math., Warsaw (1999) 193–217
  • K P Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc. 269 (1982) 351–382
  • D Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985) 1–41
  • H Sasahira, Gluing formula for the stable cohomotopy version of Seiberg–Witten invariants along $3$–manifolds with $b_1 > 0$, preprint (2014)
  • E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769–796