## Algebraic & Geometric Topology

### $\mathrm{Pin}(2)$–equivariant KO–theory and intersection forms of spin $4$–manifolds

Jianfeng Lin

#### Abstract

Using the Seiberg–Witten Floer spectrum and $Pin(2)$–equivariant $KO$–theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$–spheres. We then give explicit constrains on the intersection forms of spin $4$–manifolds bounded by Brieskorn spheres $± Σ(2,3,6k ± 1)$. Along the way, we also give an alternative proof of Furuta’s improvement of $10 8$–theorem for closed spin $4$–manifolds.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 2 (2015), 863-902.

Dates
Revised: 4 July 2014
Accepted: 3 August 2014
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895792

Digital Object Identifier
doi:10.2140/agt.2015.15.863

Mathematical Reviews number (MathSciNet)
MR3342679

#### Citation

Lin, Jianfeng. $\mathrm{Pin}(2)$–equivariant KO–theory and intersection forms of spin $4$–manifolds. Algebr. Geom. Topol. 15 (2015), no. 2, 863--902. doi:10.2140/agt.2015.15.863. https://projecteuclid.org/euclid.agt/1511895792

#### 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, Berlin (1984) 483–532
• M,F Atiyah, $K\!$–theory, W,A Benjamin, New York
• M,F Atiyah, $K\!$–theory and reality, Quart. J. Math. Oxford Ser. 17 (1966) 367–386
• M,F Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. 19 (1968) 113–140
• M,F Atiyah, G,B Segal, Equivariant $K\!$–theory and completion, J. Differential Geometry 3 (1969) 1–18
• T tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics 766, Springer, Berlin (1979)
• T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co., Berlin (1987)
• S,K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
• S,K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
• K,A Frøyshov, The Seiberg–Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996) 373–390
• K,A Frøyshov, Equivariant aspects of Yang–Mills Floer theory, Topology 41 (2002) 525–552
• K,A Frøyshov, Monopole Floer homology for rational homology $3$–spheres, Duke Math. J. 155 (2010) 519–576
• Y Fukumoto, M Furuta, Homology $3$–spheres bounding acyclic $4$–manifolds, Math. Res. Lett. 7 (2000) 757–766
• M Furuta, Monopole equation and the $\frac{11}8$–conjecture, Math. Res. Lett. 8 (2001) 279–291
• M Furuta, Y Kametani, Equivariant maps between sphere bundles over tori and $\mathrm{KO}$–degree
• M Furuta, T-J Li, Intersection forms of spin $4$–manifolds with boundary, preprint (2013)
• P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007) 457–546
• C Manolescu, $\mathrm{Pin}(2)$–equivariant Seiberg–Witten Floer homology and the triangulation conjecture
• C Manolescu, Seiberg–Witten–Floer stable homotopy type of three-manifolds with $b\sb 1=0$, Geom. Topol. 7 (2003) 889–932
• C Manolescu, A gluing theorem for the relative Bauer–Furuta invariants, J. Differential Geom. 76 (2007) 117–153
• C Manolescu, On the intersection forms of spin four-manifolds with boundary, Math. Ann. 359 (2014) 695–728
• Y Matsumoto, On the bounding genus of homology $3$–spheres, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 287–318
• N Minami, The $G$–join theorem: An unbased $G$–Freudenthal theorem, preprint
• W,D Neumann, An invariant of plumbed homology spheres, from: “Topology Symposium”, (U Koschorke, W,D Neumann, editors), Lecture Notes in Math. 788, Springer, Berlin (1980) 125–144
• W,D Neumann, F Raymond, Seifert manifolds, plumbing, $\mu$–invariant and orientation reversing maps, from: “Algebraic and geometric topology”, (K,C Millett, editor), Lecture Notes in Math. 664, Springer, Berlin (1978) 163–196
• P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
• V,A Rokhlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR 84 (1952) 221–224 In Russian
• N Saveliev, Fukumoto–Furuta invariants of plumbed homology $3$–spheres, Pacific J. Math. 205 (2002) 465–490
• B Schmidt, Spin $4$–manifolds and $\mathrm{Pin}(2)$–equivariant homotopy theory, PhD thesis, Universität Bielefield (2003) Available at \setbox0\makeatletter\@url http://pub.uni-bielefeld.de/publication/2305403 {\unhbox0
• G Segal, Equivariant $K\!$–theory, Inst. Hautes Études Sci. Publ. Math. (1968) 129–151
• L Siebenmann, On vanishing of the Rokhlin invariant and nonfinitely amphicheiral homology $3$–spheres, from: “Topology Symposium”, (U Koschorke, W,D Neumann, editors), Lecture Notes in Math. 788, Springer, Berlin (1980) 172–222
• S Stolz, The level of real projective spaces, Comment. Math. Helv. 64 (1989) 661–674