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Sparseness for the symmetric hit problem at all primes

Published online by Cambridge University Press:  11 December 2014

DAVID PENGELLEY  and
FRANK WILLIAMS
DAVID PENGELLEY
Affiliation:
Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, U.S.A. e-mail: davidp@nmsu.edu, frank@nmsu.edu
FRANK WILLIAMS
Affiliation:
Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, U.S.A. e-mail: davidp@nmsu.edu, frank@nmsu.edu
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Abstract

The hit problem for a module over the Steenrod algebra $\mathcal{A}$ seeks a minimal set of $\mathcal{A}$-generators (“non-hit elements,” those not in the image of positive degree elements of $\mathcal{A}$). This problem has been studied for 25 years in a variety of contexts and, although general or complete results have been difficult to obtain, partial results have been obtained in many cases.

For any prime p ⩾ 2, consider the algebra of symmetric polynomials in l variables over $\mathbb{F}$p (the cohomology of BU(l) [BO(l) for p = 2]). We prove a general sparseness result: For any l, the $\mathcal{A}$-generators can be concentrated in complex [real for p = 2] degrees τ such that α((p - 1) (τ + l)) ⩽(p - 1)l, where α(m) denotes the sum of the digits in the p-ary expansion of m.

The specialisation of this result to p = 2 was proved by Janfada and Wood using different methods. Key ingredients of our approach are an action of the Kudo–Araki–May algebra on the $\mathcal{A}$-primitives in homology, and a symmetric homology adaptation of the concept behind the χ-trick in cohomology.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 269 - 274
Copyright
Copyright © Cambridge Philosophical Society 2014 

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