Speaker
Description
Classical results in equivariant topology (e.g., Borsuk--Ulam theorem, $\mathbb{Z}_p$-Borsuk--Ulam theorem etc.) have numerous important applications in combinatorics. In this paper, we prove a Hopf--trace type formula, which is a purely combinatorial identity, involving no homology. This theorem provides a unified combinatorial framework for several results in equivariant topology. In fact, we use the Hopf--trace type formula to prove several combinatorial Borsuk--Ulam type theorems, e.g., $\mathbb{Z}_p$-Tucker's lemma, combinatorial degree version of $\mathbb{Z}_p$-Tucker's lemma, $\mathbb{Z}_p$-Borsuk--Ulam theorems etc. Our proofs are purely combinatorial in the sense that they do not involve homology, cohomology or any other notions from continuous topology. The combinatorial degree version of $\mathbb{Z}_p$-Tucker's lemma seems to be new in the combinatorial literature.