Subdivisions and secondary invariants
Doctor of Philosophy
Combinatorial transgressions are secondary invariants of a triangulable space analogous to transgressive forms such as those arising in Chern-Weil theory. Unlike combinatorial characteristic classes, combinatorial transgressions have not been previously studied. I characterize transgressions that are path-independent of subdivision sequence and demonstrate a canonical local formula for a particular example: namely, the difference of Poincare duals to the Euler class. Also, I show how differences in harmonic cycles are useful in examining subdivisions. In particular, I provide an algorithmically computable quantity which measures the complexity of subdivisions and may be helpful in determining the stellarity of a subdivision. An answer to this question without recourse to combinatorial exhaustion is a classical open problem.