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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
n-component link. In earlier work a recursive atom ordering was used to compute
these groups' cohomology with group ring coefficients [1]. While it is a specula-
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