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SUMMARY:A spectrum-level splitting of the $ku_\\mathbb{R}$-cooperations al
 gebra - Sarah Petersen (University of Colorado Boulder)
DTSTART:20250619T130000Z
DTEND:20250619T140000Z
UID:TALK230710@talks.cam.ac.uk
DESCRIPTION:In the 1980's\, Mahowald and Kane used integral Brown-Gitler s
 pectra to decompose $ku \\wedge ku$ as a sum of finitely generated $ku$-mo
 dule spectra. This splitting\, along with an analogous decomposition of $k
 o \\wedge ko\,$ led to a great deal of progress in stable homotopy computa
 tions and understanding of $v_1$-periodicity in the stable homotopy groups
  of spheres. In this talk\, we construct a $C_2$-equivariant lift of Mahow
 ald and Kane's splitting of $ku \\wedge ku$. We also describe the resultin
 g $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers a
 nd record an analogous splitting for $H \\underline{\\mathbb{Z}} \\wedge H
  \\underline{\\mathbb{Z}}$. Similarly to the nonequivariant story\, we out
 line how these techniques facilitate further $C_2$-equivariant stable homo
 topy computations and understanding of $v_1$-periodicity in $C_2$-equivari
 ant stable stems.
LOCATION:Seminar Room 1\, Newton Institute
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