Counting homotopy classes of mappings via Dijkgraaf-Witten invariants

摘要

Suppose Γ is a finite group acting freely on S~n (n ≥ 3 being odd) and M is any closed oriented n-manifold. We show that, given an integer k, the set deg~(-1)(k) of based homotopy classes of mappings with degree k is finite and its cardinality depends only on the congruence class of k modulo #Γ; moreover, #deg~(-1)(k) can be expressed in terms of the Dijkgraaf-Witten invariants of M.