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THE HYPERTREE POSET AND THE # 2 BETTI NUMBERS OF THE MOTION GROUP OF THE TRIVIAL LINK
 

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.

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics