Topology and group theory have enjoyed a symbiotic relationship since the late nineteenth century with particularly close ties being affirmed and reformed in recent decades. Bogley's research focuses on geometric and homotopical properties of cell complexes and their applications in group theory. Bogley's published works include results in these areas:

1. Asphericity of cell complexes and cohomology of groups
2. Relative homotopy groups and equations over groups
3. Actions on cell complexes and finiteness properties of groups
4. Combinatorial geometry and algorithmic/decidability properties of groups
5. Wild metric cell complexes and omega-group structures, in which certain infinite products can be formed subject to suitable generalizations of the associativity axiom.