From: "Samson Saneblidze"
Date: Wed, 2 Aug 2006 21:02:26 +0400
>> Subject: Massey products
>> From: Mark Grant
>> Date: Tue, 1 Aug 2006 17:58:50 +0100 (BST)
>>
>> Hi everyone,
>> does anyone know about the behaviour of Massey triple products (with
>> field coefficients) with respect to external products? I had hoped for
>> something simple like
>>
>> < a\otimes b,c\otimes d,e\otimes f>\subseteq\otimes**
>>
>> (apologies for the tex speak) but it seems this is either false or
>> tricky to prove (involving Eilenberg-Zilber maps and wotnot).
A correct fact on a behavior of Massey products with respect to the tensor
products is the
following: If both Massey triple products
< a ,c, e > and < b, d, f > are defined, then
the Massey triple product < a \otimes b, c \otimes d, e \otimes f >
is
defined and
contains zero .
Note that the analogous statement is valid for Massey products of any
order as well.
For example, such a behavior of Massey products on tensor products can be
immediately
deduced by combining the following two facts: 1. the relationship between
Massey
products and A(infty)-algebra operations on the homology of any dg
algebra;
2. the
explicit formula expressing the A(infty)-algebra operations on the tensor
product by
the operations on tensor factors (for the second item, see "S. Saneblidze
and R.
Umble, Diagonals on the permutahedra, multiplihedra and associahedra,
Homotopy,
Homology and Appl., 6(1), (2004), 363-411, math. AT/0209109 ").
Samson Saneblidze
**