  We study a variety of natural constructions from topological combinatorics,
including matching complexes as well as other graph complexes, from the
perspective of the graph minor category of \parencite{MiProRa}. We prove that
these complexes must have universally bounded torsion in their homology across
all graphs of bounded genus. One may think of these results as arising from an
algebraic version of the graph minor theorem of Robertson and Seymour
\parencite{RSXX,RSXXIII}.
