Subject: Kunneth in general
From: jim stasheff
Date: Tue, 21 Nov 2006 10:51:26 -0500
The Kunneth theorem holds for products of simplicial sets
As i recall, it does not hold for cubical sets
but does if we normalize
Can anyone supply a reference for a proof of that?
Is there some general result whihc identifies
when something like simplicial or cubical
responds similarly?
e.g. associahedral sets or Saneblidze's cubical sets?
jim
Subject: Re: Kunneth in general
From: "Samson Saneblidze"
Date: Tue, 21 Nov 2006 22:15:54 +0400
Dear Jim,
If I correctly understand the question an answer is the following: If $X$
and $Y$ are
two abstract cubical sets then the set-theoretical cartesian product
$X\times Y$
admits a canonical cubical set structure such that $C(X\times
Y)=C(X)\otimes
C(Y)$ (I
think first such a cartesian product is considered in D. M. Kan, Abstract
homotopy I,
Proc. Nat. Acad. Sci. U.S.A., 41 (1955), 1092-1096).
So that the Kunneth theorem
holds without any normalization (independently from the Kunneth theorem
the
normalization is needed for the singular cubical set to give the singular
(simplicial)
(co)homology of a topological space; but here it is enough to normalize
only
by one
degeneracy operator, as, for example, Serre(1950) did).
In general, if $X$ and $Y$ were some abstract Z-sets the above question
has a sense if
first one can introduce on the set-theoretical cartesian product $X\times
Y$
the 'same'
structure. Then if $Z$ is an acyclic polytope, the acyclic model theorem
must give a
chain equivalence $C(X\times Y)\approx C(X)\otimes C(Y),$ and,
consequently, the
Kunneth theorem.
This is the case when $Z$ is a permutahedron; in particular the product of
abstract
permutahedral sets is considered in S. Saneblidze and R. Umble, Diagonals
on the
permutahedra, multiplihedra and associahedra, \textit{J. Homology,
Homotopy
and Appl}.,
{6 }(1) (2004), 363-411 (it somehow mimics Kan's cubical product set
above!).
Since the polytopes you mention are acyclic and can be viewed as obtained
from
permutahedra by cellular projections the above procedure must be valid
for
them too
(probably, without mentioning permutahedra).
Best,
Samson