Counting homotopy classes of mappings via Dijkgraaf–Witten invariants

Suppose Γ   is a finite group acting freely on SnSn (n≥3n≥3 being odd) and M is any closed oriented n-manifold. We show that, given an integer k  , the set deg−1(k)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)#deg−1(k) can be expressed in terms of the Dijkgraaf–Witten invariants of M.