Embedding up to homotopy type— The first obstruction

For a 2n?m connected map from an n-dimensional complex to a m-dimensional manifold, an obstruction to embedding up to homotopy type is defined. The vanishing of this obstruction is a necessary and sufficient condition (in the 2n?m connected case, 2n?m ? 2, m?n ?3) to obtain an embedding up to homotopy type. In case the target manifold is Euclidean space, it is shown that the obstruction vanishes if and only if certain Thom operations are trivial. A classification theorem is given in the 2n?m+1 connected case.