3-torsion in the Homology of Complexes of Graphs of Bounded Degree

• For delta >= 1 and n >= 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most delta; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When delta = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n - 4)/3 <= d <= (n - 6)/2. This paper establishes similar bounds for delta >= 2. Specifically, there is 3-torsion in degree d whenever (3 delta - 1)n - 8/6 <= d <= delta(n - 1) - 4/2. The procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3z is a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

simplicial complex

simplicial homology

torsion group

vertex degree

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