Subject: Re: 2 questions and conf (Moore conjecture) Date: Mon, 29 Oct 2001 13:30:42 -0500 From: Zbigniew Fiedorowicz To: Don Davis Assuming we are discussing the same Moore conjecture, this question was posed to this list a couple of years ago, and I repost my answer below (with some minor typos corrected): I've worked out a counterxample to the conjecture attributed to Kan-Moore, that |M_*| --> |U(M_*)| is a homotopy equivalence for M_* a simplicial monoid with \pi_0 a group. As I suspected, it follows from MacDuff's result about realizing connected CW homotopy types as classifying spaces of discrete monoids. It doesn't require any additional properties of such a construction. Start out with a discrete monoid M whose classifying space is a simply connected noncontractible space. The simplest example is the 5 element monoid consisting of 1 and elements x_{ij}, i,j=1,2 with multiplication table x_{ij}x_{kl}=x_{il}. Its classifying space is the 2-sphere. Now form the simplicial monoid M_* whose monoid of k-simplices is the k-fold free product (in the category of monoids) of M with itself. Define the 0-th and last face map to be the homorphism which kills the first, resp. last free summand, and for remaining i, let the i-th face to be the i-th codiagonal. Define the i-th degeneracy to be the inclusion which misses the i-th free summand. Now the classifying space of this simplicial monoid is the simplicial topological space whose space of k-simplices is the k-fold wedge of S^2. The first and last face drop the first and last wedge summand, whereas the middle faces are given by fold maps. The degeneraces are given by inclusions of wedge summands. Since everything in degrees >1 is degenerate, the geometric realization of this simplicial space is S(S^2)=S^3. Since M_0=1, \pi_0(M_*)=1, and it follows that the geometric realization of M_* has homotopy type \Omega(S^3). On the other hand, U(M_*) is the trivial simplicial set. Zig Fiedorowicz