Cofreeness in real bordism theory and the segal conjecture

Carrick, Christian

doi:10.1090/proc/15702

Cofreeness in real bordism theory and the segal conjecture
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by Christian Carrick PDF

Proc. Amer. Math. Soc. 150 (2022), 3161-3175 Request permission

Abstract:

We prove that the genuine $C_{2^n}$-spectrum $N_{C_{2}}^{C_{2^n}}MU_{\mathbb {R}}$ is cofree, for all $n$. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal Conjecture for $C_2$, independent of Lin’s theorem.

References

O. Antolín-Camarena and Tobias Barthel, Chromatic fracture cubes, Eprints, arXiv:1410.7271, October 2014.
C. Carrick, Smashing localizations in equivariant stable homotopy, Preprint, arXiv:1909.08771v2, 2020.
Daniel Dugger, An Atiyah-Hirzebruch spectral sequence for $KR$-theory, $K$-Theory 35 (2005), no. 3-4, 213–256 (2006). MR 2240234, DOI 10.1007/s10977-005-1552-9
J. Hahn and D. Wilson. Real topological Hochschild homology and the Segal Conjecture. arXiv preprint arXiv:1911.05687, 2019.
Lars Hesselholt and Ib Madsen, On the $K$-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29–101. MR 1410465, DOI 10.1016/0040-9383(96)00003-1
M. A. Hill, M. J. Hopkins, and D. C. Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016), no. 1, 1–262. MR 3505179, DOI 10.4007/annals.2016.184.1.1
Po Hu and Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001), no. 2, 317–399. MR 1808224, DOI 10.1016/S0040-9383(99)00065-8
W. H. Lin, D. M. Davis, M. E. Mahowald, and J. F. Adams, Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 459–469. MR 569195, DOI 10.1017/S0305004100056899
Sverre Lunøe-Nielsen and John Rognes, The topological Singer construction, Doc. Math. 17 (2012), 861–909. MR 3007679
J. Lurie, Higher algebra, Available at http://people.math.harvard.edu/~lurie/papers/HA.pdf, 2017.
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659, DOI 10.1515/9781400830558
M. A. Mandell and J. P. May, Equivariant orthogonal spectra and $S$-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 1922205, DOI 10.1090/memo/0755
L. Meier, X. D. Shi, and M. Zeng, Norms of Eilenberg-MacLane spectra and real bordism, arXiv:2008.04963, 2020.
Thomas Nikolaus and Peter Scholze, On topological cyclic homology, Acta Math. 221 (2018), no. 2, 203–409. MR 3904731, DOI 10.4310/ACTA.2018.v221.n2.a1
John Richard Ullman, On the Regular Slice Spectral Sequence, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 3211466
D. Wilson, On categories of slices, arXiv:1711.03472, 2017.