Subject: Question about a simplicial complex From: Zbigniew Fiedorowicz Date: Sun, 02 Sep 2007 22:12:01 -0400 In some recent work with my Ph.D student, Shaun Ault, we've run into the following simplicial complex, which seems so natural that we thought it might have been encountered previously in other contexts. This is a simplicial complex in the classical sense, i.e. a simplex is uniquely determined by its vertices. Let X be a finite set of cardinality p+1. The vertex set of the simplicial complex is the set of all ordered pairs (x,y)\in X\times X with x not equal to y. The simplicial complex in question is the union of (p+1)! simplices of dimension p as follows: for each total ordering (x_{i_0},x_{i_1},...,x_{i_p}) of the set X, the corresponding simplex has vertices (x_{i_0},x_{i_1}), (x_{i_1},x_{i_2}),...,(x_{i_{p-1}},x_{i_p}). The actual complex we are interested in is the suspension of this complex, which we denote Sym^{(p))_*. We would like to have a nice description of the homology of this complex. Our computations so far have given the following results. For p\leq 7, H_*(Sym^{(p))_*) is free abelian with the following Poincare polynomials: 1 for p=0 t for p=1 t+2t^2 for p=2 7t^2+6t^3 for p=3 43t^3+24t^4 for p=4 t^3+272t^4+120t^5 for p=5 36t^4+1847t^5+720t^6 for p=6 829t^5+13710t^6+5040t^7 for p=7 In all these cases Sym^{(p))_* has the homotopy type of a wedge of spheres. We can show in general that H_p(Sym^{(p))_*) is free abelian of rank p! and that this top-dimensional homology splits off as a wedge of p-spheres. Also the Euler characteristic of Sym^{(p))_* is the numerator of the coefficient of x^{p+1} of the power series expansion of exp(x/(1+x)). Some conjectures that seem natural is that H_*(Sym^{(p))_*) is free abelian for all p and perhaps that Sym^{(p))_* decomposes as a wedge of spheres. Also that Sym^{(p))_* becomes more and more highly connected as p increases (roughly p/2 connected). For some context as to how we became interested in this simplicial complex, you might take a look at our preprint on Briefly we are looking at an analog of cyclic homology, where the cyclic groups are replaced by symmetric groups. We call this analog symmetric homology and have shown that it is strongly related to stable homotopy theory. My question to this list is if anyone has seen this or similar simplicial complexes in their work? Also if anyone has helpful suggestions, we would be very grateful.