IMBEDDING OF MANIFOLDS IN EUCLIDEAN SPACE

Abstract

This chapter discusses the piecewise linear imbeddings in Rq of compact, n-dimensional, combinatorial manifolds that are (m – 1)-connected, where 0<2m ≤ n. The condition m > 0 means that such a manifold is connected. If a closed, that is, compact, unbounded, n-manifold M is (m – l)-connected and 2m >n, then it follows from the Poincare duality that M has the homotopy type of a w-sphere. Therefore, if it turns out that every such manifold is a combinatorial n-sphere, or even if it can be piecewise linearly imbedded in Rn+1, then the theorem proved in the chapter is valid for 0<m ≤ n. The chapter presents the proof of the theorem that states that if 0<2m ≤ n, then every closed, (m – l)-connected n-manifold can be imbedded in R2n–m+1.