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Symmetric operations for all primes and Steenrod operations in algebraic cobordism

Part of: $K$-theory in geometry Differential topology Homology and cohomology theories

Published online by Cambridge University Press:  22 December 2015

Alexander Vishik
Alexander Vishik*
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email [email protected]
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Abstract

In this article we construct symmetric operations for all primes (previously known only for $p=2$). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all $p$-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map $\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$, providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.

Keywords

algebraic cobordismsymmetric operationsSteenrod operations

MSC classification

Primary: 19E15: Algebraic cycles and motivic cohomology 55N22: Bordism and cobordism theories, formal group laws 57R77: Complex cobordism (${rm U}$- and ${rm SU}$-cobordism)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 1052 - 1070
Copyright
© The Author 2015 

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