Invariant theory and the lambda algebra

Author:
William M. Singer

Journal:
Trans. Amer. Math. Soc. **280** (1983), 673-693

MSC:
Primary 55Q45; Secondary 55S10, 55T15, 55U10

DOI:
https://doi.org/10.1090/S0002-9947-1983-0716844-7

MathSciNet review:
716844

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $A$ be the Steenrod algebra over the field ${F_2}$. In this paper we construct for any left $A$-module $M$ a chain complex whose homology groups are isomorphic to the groups $\operatorname {Tor}_s^A({F_2},M)$. This chain complex in homological degree $s$ is built from a ring of invariants associated with the action of the linear group $G{L_s}({F_2})$ on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of $G{L_2}({F_2})$. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.

- A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger,
*The ${\rm mod}-p$ lower central series and the Adams spectral sequence*, Topology**5**(1966), 331โ342. MR**199862**, DOI https://doi.org/10.1016/0040-9383%2866%2990024-3 - A. K. Bousfield and E. B. Curtis,
*A spectral sequence for the homotopy of nice spaces*, Trans. Amer. Math. Soc.**151**(1970), 457โ479. MR**267586**, DOI https://doi.org/10.1090/S0002-9947-1970-0267586-7 - Leonard Eugene Dickson,
*A fundamental system of invariants of the general modular linear group with a solution of the form problem*, Trans. Amer. Math. Soc.**12**(1911), no. 1, 75โ98. MR**1500882**, DOI https://doi.org/10.1090/S0002-9947-1911-1500882-4 - Leif Kristensen,
*On a Cartan formula for secondary cohomology operations*, Math. Scand.**16**(1965), 97โ115. MR**196740**, DOI https://doi.org/10.7146/math.scand.a-10751 - Ib Madsen,
*On the action of the Dyer-Lashof algebra in $H_{\ast }(G)$*, Pacific J. Math.**60**(1975), no. 1, 235โ275. MR**388392** - John W. Milnor and John C. Moore,
*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211โ264. MR**174052**, DOI https://doi.org/10.2307/1970615 - Huแปณnh Mui,
*Modular invariant theory and cohomology algebras of symmetric groups*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**22**(1975), no. 3, 319โ369. MR**422451** - William M. Singer,
*A new chain complex for the homology of the Steenrod algebra*, Math. Proc. Cambridge Philos. Soc.**90**(1981), no. 2, 279โ292. MR**620738**, DOI https://doi.org/10.1017/S0305004100058746 - William M. Singer,
*Iterated loop functors and the homology of the Steenrod algebra. II. A chain complex for $\Omega ^{k}_{s}M$*, J. Pure Appl. Algebra**16**(1980), no. 1, 85โ97. MR**549706**, DOI https://doi.org/10.1016/0022-4049%2880%2990044-4 - William M. Singer,
*The construction of certain algebras over the Steenrod algebra*, J. Pure Appl. Algebra**11**(1977/78), no. 1-3, 53โ59. MR**467746**, DOI https://doi.org/10.1016/0022-4049%2877%2990039-1 - Clarence Wilkerson,
*Classifying spaces, Steenrod operations and algebraic closure*, Topology**16**(1977), no. 3, 227โ237. MR**442932**, DOI https://doi.org/10.1016/0040-9383%2877%2990003-9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55Q45,
55S10,
55T15,
55U10

Retrieve articles in all journals with MSC: 55Q45, 55S10, 55T15, 55U10

Additional Information

Keywords:
Adams spectral sequence,
homotopy groups of spheres,
lower central series,
lambda algebra,
cohomology of the Steenrod algebra

Article copyright:
© Copyright 1983
American Mathematical Society