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 arxiv.org:
http://arxiv.org/abs/0708.1575
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.