The authors study the map \([X,S^n] \to \Hom (E(X)/ \text{Tors}, E(S^n)/\text{Tors}) \to E(S^n)/ \text{Tors} \approxeq \mathbb{Z}\) for which the first map is a Hurewicz map for a homology functor, \(E\), the second is evaluation at a fixed element and the final isomorphism is given. When \(X = T(\xi)\), the Thom space of a vector bundle, similar constructions in homotopy theory and stable homotopy lead to a subgroup of \(\mathbb{Z}\), the integers, whose generator is the codegree of \(\xi\). The authors prove a number of non-vanishing theorems for the codegree of \(\xi\) and relate their results to those of [\textit{H. J. Marcum} and \textit{D. Randall}, Proc. Am. Math. Soc. 80, 353-358 (1980; Zbl 0459.55010)].