Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-18T18:03:26.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Central Cohomology Operations and K-Theory

Published online by Cambridge University Press:  16 April 2014

Imma Gálvez-Carrillo  and
Sarah Whitehouse
Imma Gálvez-Carrillo
Affiliation:
Departament de Matemática Aplicada III, Escola D'Enginyeria de Terrassa, Universitat Politècnica de Catalunya, C. Colom 1, 08222 Terrassa, Spain, (xlink:href="m.immaculada.galvez@upc.edu">m.immaculada.galvez@upc.edu)
Sarah Whitehouse
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK, (xlink:href="s.whitehouse@sheffield.ac.uk">s.whitehouse@sheffield.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

For stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP〈n⌰ for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.

Keywords

K-theoryoperationscobordismPrimary 55S25Secondary 55N2219L41
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 699 - 711
Copyright
Copyright © Edinburgh Mathematical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adams, J. F., Lectures on generalised cohomology, in Category theory, homology theory and their applications, III, Lecture Notes in Mathematics (Springer, 1969).Google Scholar
2.Boardman, M. J., Johnson, D. C. and Wilson, W. S., Unstable operations in generalized cohomology, in Handbook of algebraic topology, pp. 687–828 (North-Holland, Amsterdam, 1995).Google Scholar
3.Galvez, I. and Whitehouse, S., Infinite sums of Adams operations and cobordism, Math. Z. 251 (2005), 475–489.Google Scholar
4.Johnson, D. C. and Wilson, W. S., Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353.Google Scholar
5.Strong, M.-J. and Whitehouse, S., Infinite sums of unstable Adams operations and cobordism, J. Pure Appl. Alg. 214 (2010), 910–918.Google Scholar
6.Strong, M.-J. and Whitehouse, S., Integer-valued polynomials and K-theory operations, Proc. Am. Math. Soc. 138(6) (2010), 2221–2233.CrossRefGoogle Scholar
7.Wilson, W. S., The Ω-spectrum for Brown-Peterson cohomology, II, Am. J. Math. 97 (1975), 101–123.CrossRefGoogle Scholar