Algebraic & Geometric Topology

The connective real $K$–theory of Brown–Gitler spectra

Paul Thomas Pearson

Full-text: Open access

Abstract

We calculate the connective real K–theory homology of the mod 2 Brown–Gitler spectra. We use this calculation and the theory of Dieudonné rings and Hopf rings to determine the mod 2 homology of the spaces in the connective Ω–spectrum for topological real K–theory.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 597-625.

Dates
Received: 30 October 2011
Revised: 5 August 2013
Accepted: 27 August 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715812

Digital Object Identifier
doi:10.2140/agt.2014.14.597

Mathematical Reviews number (MathSciNet)
MR3158769

Zentralblatt MATH identifier
1284.55008

Subjects
Primary: 55T25: Generalized cohomology
Secondary: 55P42: Stable homotopy theory, spectra

Keywords
Hopf ring Dieudonné ring topological real $K$–theory

Citation

Pearson, Paul Thomas. The connective real $K$–theory of Brown–Gitler spectra. Algebr. Geom. Topol. 14 (2014), no. 1, 597--625. doi:10.2140/agt.2014.14.597. https://projecteuclid.org/euclid.agt/1513715812


Export citation

References

  • J F Adams, A periodicity theorem in homological algebra, Proc. Cambridge Philos. Soc. 62 (1966) 365–377
  • J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press (1995)
  • A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, J W Schlesinger, The $\mathrm{mod}$–$\!p$ lower central series and the Adams spectral sequence, Topology 5 (1966) 331–342
  • E H Brown, Jr, S Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973) 283–295
  • R R Bruner, J P May, J E McClure, M Steinberger, $H\sb \infty $ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer, Berlin (1986)
  • V Buchstaber, A Lazarev, Dieudonné modules and $p$–divisible groups associated with Morava $K$–theory of Eilenberg–Mac Lane spaces, Algebr. Geom. Topol. 7 (2007) 529–564
  • R L Cohen, Odd primary infinite families in stable homotopy theory, Mem. Amer. Math. Soc. 242, Amer. Math. Soc. (1981)
  • D M Davis, S Gitler, M Mahowald, The stable geometric dimension of vector bundles over real projective spaces, Trans. Amer. Math. Soc. 268 (1981) 39–61
  • P Goerss, Hopf rings, Dieudonné modules, and $E\sb *\Omega\sp 2S\sp 3$, from: “Homotopy invariant algebraic structures”, (J-P Meyer, J Morava, W S Wilson, editors), Contemp. Math. 239, Amer. Math. Soc. (1999) 115–174
  • P Goerss, J D S Jones, M Mahowald, Some generalized Brown–Gitler spectra, Trans. Amer. Math. Soc. 294 (1986) 113–132
  • P Goerss, J Lannes, F Morel, Hopf algebras, Witt vectors, and Brown–Gitler spectra, from: “Algebraic topology”, (M C Tangora, editor), Contemp. Math. 146, Amer. Math. Soc. (1993) 111–128
  • J R Hunton, P R Turner, Coalgebraic algebra, J. Pure Appl. Algebra 129 (1998) 297–313
  • N Kitchloo, G Laures, W S Wilson, The Morava $K$–theory of spaces related to $BO$, Adv. Math. 189 (2004) 192–236
  • M Mahowald, A new infinite family in ${}_{2}\pi_{*}{}^s$, Topology 16 (1977) 249–256
  • M Mahowald, $b{\rm o}$–resolutions, Pacific J. Math. 92 (1981) 365–383
  • H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984) 39–87
  • J Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958) 150–171
  • D S C Morton, The homology of the spectrum $bo$ and its connective covers, PhD thesis, Johns Hopkins Univ. (1997) Available at \setbox0\makeatletter\@url http://www.math.jhu.edu/~wsw/papers/dena.dvi {\unhbox0
  • D S C Morton, The Hopf ring for $b{\rm o}$ and its connective covers, J. Pure Appl. Algebra 210 (2007) 219–247
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 2nd edition, AMS Chelsea Series 347, Amer. Math. Soc. (2003)
  • D C Ravenel, W S Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1976/77) 241–280
  • C Schoeller, Étude de la catégorie des algèbres de Hopf commutatives connexes sur un corps, Manuscripta Math. 3 (1970) 133–155
  • D H Shimamoto, An integral version of the Brown–Gitler spectrum, Trans. Amer. Math. Soc. 283 (1984) 383–421
  • N P Strickland, Bott periodicity and Hopf rings, PhD thesis, University of Manchester (1993) Available at \setbox0\makeatletter\@url http://neil-strickland.staff.shef.ac.uk/papers/thesis.dvi {\unhbox0
  • W S Wilson, Hopf rings in algebraic topology, Expo. Math. 18 (2000) 369–388