Floer homotopy type

ホモトピー論の視点からは, Floer homology の元になる “Floer homotopy type” があると考えるのは自然である。

実際, 様々な人々がそれを実現するアイデアを考えている。 Lipschitz と Sarkar の Khovanov homotopy type に関する survey [LS] では, 次のような仕事が紹介されている。

• Ralph Cohen と John Jones と Graeme Segal の [CJS95] とその続編 [Coh10; Coh09]
• Furuta の [Fur01]
• Bauer と Furuta の [BF04; Bau04]
• Manolescu の [Man03; Man07]
• Kronheimer と Manolescu の [KM]
• C.L. Douglas の [Dou]
• Abouzaid と Kragh の [AK]
• Kragh の [Kra; Kra13]
• Khandhawit [Khaa; Khab; KLS]
• Sasahira の [Sas]

これらの結果から, Floer homotopy type は stable homotopy type として表すべきもののように思える。

一方で, Barraud の [Bar18] では, symplectic manifold の基本群を Floer theoretic object で表すということが考えられている。

• Floer fundamental group

References

[AK]

Mohammed Abouzaid and Thomas Kragh. On the immersion classes of nearby Lagrangians. arXiv: 1305.6810.

[Bar18]

Jean-François Barraud. “A Floer fundamental group”. In: Ann. Sci. Éc. Norm. Supér. (4) 51.3 (2018), pp. 773–809. arXiv: 1404.3266. url: https://doi.org/10.24033/asens.2366.

[Bau04]

Stefan Bauer. “A stable cohomotopy refinement of Seiberg-Witten invariants. II”. In: Invent. Math. 155.1 (2004), pp. 21–40. arXiv: math/0204267. url: http://dx.doi.org/10.1007/s00222-003-0289-4.

[BF04]

Stefan Bauer and Mikio Furuta. “A stable cohomotopy refinement of Seiberg-Witten invariants. I”. In: Invent. Math. 155.1 (2004), pp. 1–19. arXiv: math/0204340. url: http://dx.doi.org/10.1007/s00222-003-0288-5.

[CJS95]

R. L. Cohen, J. D. S. Jones, and G. B. Segal. “Floer’s infinite-dimensional Morse theory and homotopy theory”. In: The Floer memorial volume. Vol. 133. Progr. Math. Basel: Birkhäuser, 1995, pp. 297–325.

[Coh09]

Ralph L. Cohen. “Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners”. In: Algebraic topology. Vol. 4. Abel Symp. Berlin: Springer, 2009, pp. 39–59. arXiv: 0802.2752. url: http://dx.doi.org/10.1007/978-3-642-01200-6_3.

[Coh10]

Ralph L. Cohen. “The Floer homotopy type of the cotangent bundle”. In: Pure Appl. Math. Q. 6.2, Special Issue: In honor of Michael Atiyah and Isadore Singer (2010), pp. 391–438. arXiv: math/0702852.

[Dou]

Christopher L. Douglas. Twisted Parametrized Stable Homotopy Theory. arXiv: math/0508070.

[Fur01]

M. Furuta. “Monopole equation and the \(\frac{11}8\)-conjecture”. In: Math. Res. Lett. 8.3 (2001), pp. 279–291. url: https://doi.org/10.4310/MRL.2001.v8.n3.a5.

[Khaa]

Tirasan Khandhawit. A new gauge slice for the relative Bauer-Furuta invariants. arXiv: 1401.7590.

[Khab]

Tirasan Khandhawit. On the stable Conley index in Hilbert spaces. arXiv: 1402.1665.

[KLS]

Tirasan Khandhawit, Jianfeng Lin, and Hirofumi Sasahira. Unfolded Seiberg-Witten Floer spectra, I: Definition and invariance. arXiv: 1604.08240.

[KM]

Peter B. Kronheimer and Ciprian Manolescu. Periodic Floer pro-spectra from the Seiberg-Witten equations. arXiv: math/0203243.

[Kra]

Thomas Kragh. The Viterbo Transfer as a Map of Spectra. arXiv: 0712.2533.

[Kra13]

Thomas Kragh. “Parametrized ring-spectra and the nearby Lagrangian conjecture”. In: Geom. Topol. 17.2 (2013). With an appendix by Mohammed Abouzaid, pp. 639–731. arXiv: 1107.4674. url: http://dx.doi.org/10.2140/gt.2013.17.639.

[LS]

Robert Lipshitz and Sucharit Sarkar. Spatial refinements and Khovanov homology. arXiv: 1709.03602.

[Man03]

Ciprian Manolescu. “Seiberg-Witten-Floer stable homotopy type of three-manifolds with \(b_1=0\)”. In: Geom. Topol. 7 (2003), pp. 889–932. arXiv: math/0104024. url: https://doi.org/10.2140/gt.2003.7.889.

[Man07]

Ciprian Manolescu. “A gluing theorem for the relative Bauer-Furuta invariants”. In: J. Differential Geom. 76.1 (2007), pp. 117–153. arXiv: math/0311342. url: http://projecteuclid.org/euclid.jdg/1180135667.

[Sas]

H. Sasahira. Gluing formula for the stable cohomotopy version of Seiberg-Witten invariants along 3-manifolds with \(b_1 > 0\). arXiv: 1408.2623.