Massey Product Shuffling Result

[Mike Hill]

I know that I have spoken with some of you about this, but I thought
others might benefit from it. Here goes.

Let a,b,c,d,e be elements in a DGA such that the Massey products
<a,b,<c,d,e>>, <a,<b,c,d>,e> and <<a,b,c>,d,e> all make sense. Then
there is a [graded] Jacobi identity:

<a,b,<c,d,e>>+<a,<b,c,d>,e>+<<a,b,c>,d,e>=0

Chris & I verified this in the case of no indeterminacy and over
characteristic 2. I'll try to work out the remaining cases later. In a
great many situations, some of these will just be zero, making
computations a lot easier.

I should also point out that a similar result holds for 4-fold Massey
products, where one of the elements is a 3-fold Massey product:

If <<a,b,c>,d,e,f> etc are all defined, then
<a,<b,c,d>,e,f>+<a,b,<c,d,e>,f>=0 (the other two terms are
automatically zero, to make the 4-fold products make sense).

M

[Mike Hill]

For those still curious, the full graded form for the nested triple
products result is:

(-1)^|a|<<a,b,c>,d,e>+<a,<b,c,d>,e>+(-1)^|b|<a,b,<c,d,e>> = 0.

This actually does have applications. With such a shuffling result, one
can immediately see the following Toda bracket in the stable stem at an
odd prime:

<p,\alpha_2,\alpha_2>=\alpha_4.