Notes on enumerating embeddings of unorientable n-manifolds in Euclidean (2n-1)-space for odd n

Denote by [M ⊂ R^m] the set of isotopy classes of embeddings of an n-manifold M in Euclidean m-space. In topology, the computation of this set is an interesting subject. The set [M ⊂ R^{2n-1}] has been studied when n is even or M is orientable [15]. Hence, in this article, we shall study the set [M ⊂ R^{2n-1}] for an n-manifold M for which n is odd and M is unorientable. Further we compute [P(m, n) ⊂ R^{2m+4n-1}] for the Dold manifold of type (m, n) of dimension m+2n, both m and n being odd