A new gauge slice for the relative Bauer–Furuta invariants

Khandhawit, Tirasan¹

¹ Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, 5-1-5 Kashiwa-No-Ha, Kashiwa, Chiba 277-8583, Japan

Geometry & topology, Tome 19 (2015) no. 3, pp. 1631-1655.

Résumé

In this paper, we study Manolescu’s construction of the relative Bauer–Furuta invariants arising from the Seiberg–Witten equations on $4$–manifolds with boundary. The main goal of this paper is to introduce a new gauge fixing condition in order to apply the finite-dimensional approximation technique. We also hope to provide a framework to extend Manolescu’s construction to general $4$–manifolds.

Détail

DOI : 10.2140/gt.2015.19.1631

Classification : 57R57, 57R58
Keywords: $4$–manifolds, Seiberg–Witten equations, Bauer–Furuta invariant

Bibliographie

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

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

[3] S Bauer, M Furuta, A stable cohomotopy refinement of Seiberg–Witten invariants, I, Invent. Math. 155 (2004) 1

[4] C Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Math. 38, Amer. Math. Soc. (1978)

[5] M Furuta, Monopole equation and the $\frac{11}8$–conjecture, Math. Res. Lett. 8 (2001) 279

[6] T Iwaniec, C Scott, B Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. 177 (1999) 37

[7] T Khandhawit, Twisted Manolescu–Floer spectra for Seiberg–Witten, PhD thesis, Massachusetts Institute of Technology (2013)

[8] P Kronheimer, C Manolescu, Periodic Floer pro-spectra from the Seiberg–Witten equations,

[9] P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Math. Monographs 10, Cambridge Univ. Press (2007)

[10] H L Kurland, On the two definitions of the Conley index, Proc. Amer. Math. Soc. 106 (1989) 1117

[11] F Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture,

[12] J Lin, $\mathrm{Pin}(2)$–equivariant KO–theory and intersection forms of spin four-manifolds,

[13] C Manolescu, $\mathrm{Pin(2)}$–equivariant Seiberg–Witten Floer homology and the triangulation conjecture,

[14] C Manolescu, Seiberg–Witten–Floer stable homotopy type of three-manifolds with $b_1=0$, Geom. Topol. 7 (2003) 889

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

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

[17] G Schwarz, Hodge decomposition: A method for solving boundary value problems, Lecture Notes in Math. 1607, Springer (1995)