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Intersection forms of spin 4-manifolds and the Pin(2)-equivariant Mahowald invariant

MICHAEL J. HOPKINS Harvard University 林剑锋 清华大学丘成桐数学科学中心 XiaoLin Danny Shi University of Chicago Zhouli Xu UC San Diego

Geometric Analysis and Geometric Topology mathscidoc:2205.15001

Communications of the American Mathematical Society, 2022.2
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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@inproceedings{michael2022intersection,
  title={Intersection forms of spin 4-manifolds and the  Pin(2)-equivariant Mahowald invariant },
  author={MICHAEL J. HOPKINS, 林剑锋, XiaoLin Danny Shi, and Zhouli Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517170841171605244},
  booktitle={Communications of the American Mathematical Society},
  year={2022},
}
MICHAEL J. HOPKINS, 林剑锋, XiaoLin Danny Shi, and Zhouli Xu. Intersection forms of spin 4-manifolds and the Pin(2)-equivariant Mahowald invariant . 2022. In Communications of the American Mathematical Society. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517170841171605244.
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