A generalization of Whitney Lemma
Kazuaki Kobayashi · Hokkaido Mathematical Journal · 1976
\S 0. In this paper we study the elimination of the intersections of manifolds which is a generalization of WHITNEY Lemma as follows.WHITNEY Lemma {simply connected version) [see [R&S] ) : Let P^{p} , Q^{q} be a pair of connected compact locally flat submanifolds of M^{m} which are transverse, so that p+q=m.Suppose (1) p\geqq 3 , q\geqq 3 and \pi_{1}(M)=0 or(2) p\geqq 2 , q\geqq 3 and \pi_{1}(M-Q)=0 .If the intersection number of P and Q, \epsilon(P, Q) , is zero, we can ambient isotope P off Q, by an isotopy which has compact support.We work in the PL category ([Z]) throughout the paper.MAIN RESULT I (BOUNDED VERS1ON) (COROLLARY To THEOREM 1).Let P be a compact p -manifold and M be an m -manifold.Let Q be a compact q-dim.submanifold of M and f:Parrow IntM be an embedding, so that p+q=m+k.If (1) \partial P eq\phi , P is k-connected, k\leqq p-3 and f(P)\cap Q \subset f (Int P) or (2) \partial Q eq\phi , Q is k-connected, k\leqq q-3 and f(P)\cap Q\subset IntQ , thm there is an mbedding g:Parrow IntM which is ambimt isotopic to f and g(P)\cap Q=\phi .MAIN RESULT II (CLOSED VERS1ON) (THEOREM 2.) Let P, M be a con- nected closed p-and m -manifolds and Q be a connected closed q-submani- fold of M. Let f:Parrow M be an embedding and let p+q=m+k.Put N=f(P)\cap Q.