 
Summary: THE HYPERTREE POSET AND THE # 2 BETTI NUMBERS
OF THE MOTION GROUP OF THE TRIVIAL LINK
JON MCCAMMOND 1 AND JOHN MEIER 2
Abstract. We give explicit formulae for the Euler characteristic and # 2
cohomology of the group of motions of the trivial link, or isomorphically the
group of free group automorphisms that send each standard generator to a con
jugate of itself. The method is primarily combinatorial and ultimately relies
on a computation of the M˜obius function for the poset of labelled hypertrees.
1. Introduction
Classic combinatorial group theory, such as what is described in [11] or [12], uses
relatively elementary combinatorics to study infinite groups. For instance, small
cancellation theory is the study of groups with finite presentations whose associ
ated Whitehead graph has large girth. Increasingly there is a need to use more
sophisticated combinatorial arguments to establish topological properties of infi
nite groups. Here we compute the M˜obius function of a poset of labelled hypertrees
in order to explicitly describe the # 2 Betti numbers of the motion group of a trivial
ncomponent link. In earlier work a recursive atom ordering was used to compute
these groups' cohomology with group ring coe#cients [1]. While it is a specula
tive claim at this point, these two examples indicate that there may be general
applicability of enumerative combinatorics in the study of group cohomology.
