A spectrum-level splitting of the $ku_\mathbb{R}$-cooperations algebra

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• Sarah Petersen (University of Colorado Boulder)
• Thursday 19 June 2025, 14:00-15:00
• Seminar Room 1, Newton Institute.

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EHTW04 - Beyond the telescope conjecture

In the 1980’s, Mahowald and Kane used integral Brown-Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko,$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this talk, we construct a $C_2$-equivariant lift of Mahowald and Kane’s splitting of $ku \wedge ku$. We also describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H \underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Similarly to the nonequivariant story, we outline how these techniques facilitate further $C_2$-equivariant stable homotopy computations and understanding of $v_1$-periodicity in $C_2$-equivariant stable stems.

This talk is part of the Isaac Newton Institute Seminar Series series.

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