ON PIECEWISE LINEAR IMMERSIONS Reference 1. H. Rossi, Vector fields on analytic spaces, Ann. of Math. 78 (1963), 455-467. Princeton University ON PIECEWISE LINEAR IMMERSIONS MORRIS W. HIRSCH The purpose of this note is to prove an existence theorem for im- mersions of piecewise linear manifolds in Euclidean space. A more comprehensive theory of piecewise linear immersions has been worked out by Haefliger and Poenaru [l]. All maps, manifolds, microbundles, etc. are piecewise linear unless the contrary is explicitly indicated. Let M be a manifold without boundary, of dimension n. Denote the tangent microbundle of M by tm, and the trivial microbundle over M of (fibre) dimension k by e*. Let be a microbundle of dimension k such that £ is a manifold. mersion of M in Rn+k is a locally one-one map /: M—>Pn+*. An im- I say / has a normal bundle of type v if there is an immersion g: E—>Rn+k such that gi=f. (It is unknown whether / necessarily has a normal bundle, or whether all normal bundles of / are of the same type.) The converse of the following theorem is trivial. Theorem. Assume that if k = 0, then M has no compact component. There exists an immersion of M in Rn+k having a normal bundle of type v if there exists an isomorphism 4>:rm® v->e+k Proof. We may assume that i(M) is a deformation retract of the total space E of v. By Milnor [3], te \i(M) is isomorphic to tm®v; it follows from the existence of that te is trivial. According to [3] Received by the editors July 29, 1964. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use