We develop a homology of vertex groups as a tool for studying orbifolds and orbifold cobordism and its torsion. To a pair of conjugacy classes of degree- and degree- finite subgroups of and we associate the parity with which occurs up to conjugacy as a vertex group in the orbifold . This extends to a map between the vector spaces whose bases are all such conjugacy classes in and then . Using orbifold graphs, we prove is a differential and defines a homology, . We develop a map for a subcomplex of groups which admit orientation-reversing automorphisms. We then look at examples and algebraic properties of and , including that is a derivation. We prove that the natural map between the set of diffeomorphism classes of closed, locally oriented –orbifolds and maps into and that this map is onto for . We relate to orbifold cobordism and surgery and show that quotients to a map between oriented orbifold cobordism and .
"Oriented orbifold vertex groups and cobordism and an associated differential graded algebra." Algebr. Geom. Topol. 15 (1) 169 - 190, 2015. https://doi.org/10.2140/agt.2015.15.169