Invariant theory and the lambda algebra
Author:
William M. Singer
Journal:
Trans. Amer. Math. Soc. 280 (1983), 673693
MSC:
Primary 55Q45; Secondary 55S10, 55T15, 55U10
MathSciNet review:
716844
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be the Steenrod algebra over the field . In this paper we construct for any left module a chain complex whose homology groups are isomorphic to the groups . This chain complex in homological degree is built from a ring of invariants associated with the action of the linear group 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 . 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 invarianttheoretic 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 invarianttheoretic language.
 [1]
A.
K. Bousfield, E.
B. Curtis, D.
M. Kan, D.
G. Quillen, D.
L. Rector, and J.
W. Schlesinger, The 𝑚𝑜𝑑𝑝 lower
central series and the Adams spectral sequence, Topology
5 (1966), 331–342. MR 0199862
(33 #8002)
 [2]
A.
K. Bousfield and E.
B. Curtis, A spectral sequence for the homotopy
of nice spaces, Trans. Amer. Math. Soc. 151 (1970), 457–479.
MR
0267586 (42 #2488), http://dx.doi.org/10.1090/S00029947197002675867
 [3]
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,
75–98. MR
1500882, http://dx.doi.org/10.1090/S00029947191115008824
 [4]
Leif
Kristensen, On a Cartan formula for secondary cohomology
operations, Math. Scand. 16 (1965), 97–115. MR 0196740
(33 #4926)
 [5]
Ib
Madsen, On the action of the DyerLashof algebra in
𝐻_{∗}(𝐺), Pacific J. Math. 60
(1975), no. 1, 235–275. MR 0388392
(52 #9228)
 [6]
John
W. Milnor and John
C. Moore, On the structure of Hopf algebras, Ann. of Math. (2)
81 (1965), 211–264. MR 0174052
(30 #4259)
 [7]
Huỳnh
Mui, Modular invariant theory and cohomology algebras of symmetric
groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22
(1975), no. 3, 319–369. MR 0422451
(54 #10440)
 [8]
William
M. Singer, A new chain complex for the homology of the Steenrod
algebra, Math. Proc. Cambridge Philos. Soc. 90
(1981), no. 2, 279–292. MR 620738
(82k:55018), http://dx.doi.org/10.1017/S0305004100058746
 [9]
William
M. Singer, Iterated loop functors and the homology of the Steenrod
algebra. II. A chain complex for
Ω^{𝑘}_{𝑠}𝑀, J. Pure Appl. Algebra
16 (1980), no. 1, 85–97. MR 549706
(81b:55040), http://dx.doi.org/10.1016/00224049(80)900444
 [10]
William
M. Singer, The construction of certain algebras over the Steenrod
algebra, J. Pure Appl. Algebra 11 (1977/78),
no. 13, 53–59. MR 0467746
(57 #7598)
 [11]
Clarence
Wilkerson, Classifying spaces, Steenrod operations and algebraic
closure, Topology 16 (1977), no. 3,
227–237. MR 0442932
(56 #1307)
 [1]
 A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector and J. W. Schlesinger, The  lower central series and the Adams spectral sequence, Topology 5 (1966), 331342. MR 0199862 (33:8002)
 [2]
 A. K. Bousfield and E. B. Curtis. A spectral sequence for the homotopy of nice spaces, Trans. Amer. Math. Soc. 151 (1970), 457479. MR 0267586 (42:2488)
 [3]
 L. E. 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), 7598. MR 1500882
 [4]
 L. Kristensen, A Curtan formula for secondary cohomology operutions, Math. Scand. 16 (1965), 97115. MR 0196740 (33:4926)
 [5]
 I. Madsen, On the action of the DyerLashof algebra in , Pacific J. Math. 60 (1975), 235275. MR 0388392 (52:9228)
 [6]
 J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211264. MR 0174052 (30:4259)
 [7]
 H. Mui, Modular invariant theory und the cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 319369. MR 0422451 (54:10440)
 [8]
 W. M. Singer, A new chain complex for the homology of the Steenrod algebra, Proc. Cambridge Philos. Soc. 90 (1981), 279292. MR 620738 (82k:55018)
 [9]
 , Iterated loop functors and the homology of the Steenrod algebra. II: A chain complex for , J. Pure Appl. Algebra 16 (1980), 8597. MR 549706 (81b:55040)
 [10]
 , The construction of certain algebras over the Steenrod algebra, J. Pure Appl. Algebra 11 (1977), 5359. MR 0467746 (57:7598)
 [11]
 C. Wilkerson, Classifying spaces, Steenrod operations, and algebraic closure, Topology 16 (1977), 227237. MR 0442932 (56:1307)
Similar Articles
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
DOI:
http://dx.doi.org/10.1090/S00029947198307168447
PII:
S 00029947(1983)07168447
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
