Formal Groups and the Secondary Steenrod algebra
File(s): none available
Status: planning
Abstract
The goal is to give a description of H.J.Baues' "secondary Steenrod algebra" in terms of formal group laws.
Project details
Baues' secondary Steenrod algebra is a certain 4-term exact sequence
of the form
At odd primes this sequence is probably for all intents and purposes equivalent to the secondary Steenrod algebra [and I expect this to be easy to proof for those who are on proving terms with Baues' theory].
For p=2 there are extra complications which are measured by the non-triviality of certain maps S and L (the "symmetry operator" and the "left action" map). I have formulas for these maps which seem to be related to the power series
A → B1 → B0 → A
where A is the ordinary Steenrod algebra.
At odd primes it is very easy to describe a similar sequence
based on BP.
For this define
EBP = BP ⊗ E(β0, β1,...)
with ∂ βi = vi.
Let wi = ∂ (β0 βi) = p βi - vi β0.
Then there is an exact sequence
A → A + Σi≥0 βi A + Σj≥1 β0βj A → A + Σi≥0 vi A + Σj≥1 wj A → A
where the third term is interpreted as a subalgebra of
the EBP operations modulo I2.
At odd primes this sequence is probably for all intents and purposes equivalent to the secondary Steenrod algebra [and I expect this to be easy to proof for those who are on proving terms with Baues' theory].
For p=2 there are extra complications which are measured by the non-triviality of certain maps S and L (the "symmetry operator" and the "left action" map). I have formulas for these maps which seem to be related to the power series
F(x,y) = x+y+Σi,j≥0 λ i,j
x2i
y2j
This is actually the universal example of an associative, but not necessarily commutative,
formal group law over a square-zero extension of ℤ/2.
This suggests that a study of "homotopy commutative" formal group laws (over very simple rings)
could lead to a better understanding of the secondary Steenrod algebra at the prime 2.