Lin, JianfengShi, XiaoLin DannyXu, ZhouliMichael, HopkinsHopkins, Michael2023-07-252022-02-23Lin, Jianfeng, XiaoLin Danny Shi, Zhouli Xu, Hopkins Michael, Michael Hopkins. "Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant." Comm. Amer. Math. Soc. 2, no. 2 (2022): 22-132. DOI: 10.1090/cams/42692-3688https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37376598<p>In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem.</p> <p>We prove our theorem by analyzing maps between certain finite spectra arising from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>Pin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-based Atiyah–Hirzebruch spectral sequence.</p>en-USJournal Article2023-07-2510.1090/cams/4