From immersions to embeddings of smooth manifolds

Abstract

Introduction. In this paper we study the question of when an immersion between two smooth manifolds, f: V -+ M, is homotopic to an embedding. We introduce an invariant y(f) which determines, under suitable connectivity hypotheses, whether f is homotopic to an embedding or not. Specifically, if Vn is k/2 connected, Mm k-connected and m > 2n -k, our main theorem (3.5) shows thatf is homotopic to an embedding if y(f) = 0, m > (3/2)(n + 1), and the normal bundle to the immersion has a cross section. Thus the connectivity of V is about half of what is usually required. The ifvariant y(f) is usually readily computed, since there is a simple formula for it in terms of Poincare duality on V and M, the fundamental class of V, and the mapsf* andf* (see 2.8, 2.10). In particular it is always zero for M=Rm (4.2). This yields essentially the results of deSapio [2] immediately. For the standard embedding questions about CPn and HPn it yields (using 4.3): (i) If CPn immerses in R4 ,3 then it embeds in R4Th2. (ii) If Hpn immerses in R8-j', it embeds in R8n-1+1 forj< 8. Actually, the method gives more interesting results when MO Rm. For example if V and M are products of spheres (with correct connectivity) one can pretty nearly classify which continuous maps are homotopic to embeddings. Such special cases are developed at the end of Chapter 4. As for notation, v always denotes a normal bundle, Vx denotes the tangent space of Vn at x. Throughout the whole paper, iff: Vn ___ Mm is an immersion, the letter r is reserved exclusively for 2n m, the dimension of the double point set. All manifolds and immersions will be smooth (i.e. C ) unless expressly referred to as piecewise linear.