Vol. 41, No. 1, 1972

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Group representations and the Adams spectral sequence

R. James Milgram

Vol. 41 (1972), No. 1, 157–182

Homotopy groups admit primary operations analogous to the Steenrod operations in ordinary cohomology theory and a good understanding of them seems vital to interpreting patterns in the homotopy of spheres.

Also, it has been known for a long time that a type of Steenrod algebra acts in ExtA(Zp,Zp) if A is a cocommutative Hopf algebra. Recently, D. S. Kahn showed that in the E2 term of the Adams spectral sequence Ext𝒜(2)∗∗(Z2,Z2), certain of these operations on infinite cycles converge to the graded elements associated to the actual homotopy operations. Also, on infinite cycles, he showed how this structure determined some differentials.

In this paper, we further explore the relations between the operations in Ext𝒜(p)∗∗(Zp,Zp) and differentials in the Adams spectral sequence. In particular, for elements which need not be infinite cycles, we prove

Theorem 4.1.1.

(a) There are operations Sqi in Ext𝒜(2)(Z2,Z2) so that

i       h0Sqi+1(a),  i ≡ s(2)
∂2(Sq (a)) =  0           otherwise

for a Ext𝒜(2)r,s(Z2,Z2).

(b) There are operations 𝒫i𝒫i in Ext𝒜(p)(Zp,Zp) for p an odd prime so that

∂ (𝒫i(a)) = α β𝒫i(a),
2          0

for a Ext𝒜(p)r,s(Zp,Zp). (Here, Sqi takes Exts,r homomorphically to Exts+i,2r while 𝒫i takes Exts,r to Exts+(2ir)(p1),pr, and β𝒫i(Exts,r) Exts+(2ir)(p1)+1,pr.)

Mathematical Subject Classification 2000
Primary: 18H10
Secondary: 20G99, 55H15
Received: 19 October 1970
Published: 1 April 1972
R. James Milgram
Stanford University
Stanford CA 94305
United States