Smooth maps of constant rank

Introduction. In this announcement the Smale-Hirsch classification of immersions ([8], THEOREM 1. If M is open and has a proper Morse function with no critical points of index >A;, then the differential map d:Hom k (M 9 W)-+ Lin k (TM, TW) is a weak homotopy equivalence. (A manifold with such a Morse function will be said to have geometric dimension ^k. We will write geo dim M^K.) Notation. M and W are smooth manifolds with tangent bundles TM, TW; Hom k (M, W) is the space of smooth maps of rank k from M to W, with the C^compact-open topology; Lin k (TM, TW) is the space of continuous maps: TM-+TW which are fiberwise linear maps of rank k, with the compact open topology; d(f)=df REMARKS. 1. Weakening the hypotheses leads to false statements. If M is not open there are counterexamples when k=dim W as. in [7].