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On the Minimality of Anick's Resolution, with Application to an Algebra of Cohomology Operations

✍ Scribed by William M. Singer

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
167 KB
Volume
179
Category
Article
ISSN
0021-8693

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✦ Synopsis

INTRODUCTION AND STATEMENTS OF RESULTS

w x

Let G be an augmented algebra over a field k. In his paper 1 , Anick supposes given a set S of generators for G, together with a grading e: S Βͺ Z q and a total orderfor S, such that S is well ordered in each 0 Γ„ 4 degree. We will refer to the triple S, e,as ''generating data'' for G. 0 To any given generating data, Anick associates a free resolution of k as a right G-module.

Our object in this paper is to give a simple condition on the generating data that implies the associated Anick resolution is minimal. Thus, for algebras G admitting data that satisfies our condition, we determine the G Ε½ . homology groups Tor k, k for all n G 0.

w x Let H * RP s F x be the mod-2 cohomology of infinite dimensional 2 real projective space; a polynomial ring on a single generator of degree 1. w x In 2 , Giambalvo and Peterson define an interesting subalgebra, which we Ε½ Ο± . will call GP, of the ring of linear endomorphisms of H * RP . Using some w x results from 2 , we will show that GP admits generating data satisfying our criterion for the minimality of Anick's resolution. Consequently we deter-GP Ε½ . mine the homology groups Tor F , F for all n G 0. This computation n 2 2 may be of value in algebraic topology, because there is an interesting algebra homomorphism from the Steenrod algebra A into GP. We expect the ideas in this paper to be of use in applying Anick's resolution to the homology of certain subquotients of A, and possibly to the homology of A itself.

We close this introductory section by describing briefly our condition for the minimality of Anick's resolution. Suppose G is an augmented Γ„ 4 k -algebra, and let generating data S, e,be given. Let B be the free 0 monoid generated by S. Under these conditions, Anick partitions B into 918

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