Imbedding of Manifolds in Euclidean Space

Abstract

This chapter discusses the piecewise linear imbeddings in R q of compact, n -dimensional, combinatorial manifolds that are (m – 1)-connected, where 0 m ≤ 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 2 m > 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 R n+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 m ≤ n , then every closed, (m – l)-connected n-manifold can be imbedded in R 2n–m+1 .