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