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Pin(2)-equivariant KO-theory and intersection forms of spin four-manifolds

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 Added by Jianfeng Lin
 Publication date 2014
  fields
and research's language is English
 Authors Jianfeng Lin




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Using 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. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $pmSigma(2,3,6kpm1)$. Along the way, we also give an alternative proof of Furuta-Kametannis improvement of 10/8-theorem for closed spin-4 manifolds.



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In studying the 11/8-Conjecture on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a 10/8+4-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral sequence.
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