Counting homotopy classes of mappings via Dijkgraaf–Witten invariants

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.