Embeddings of k-connected n-manifolds into R^{2n-k-1}

TL;DR

This paper provides estimations for the classification of embeddings of closed k-connected n-manifolds into Euclidean space, using Whitney invariants and algebraic methods, advancing understanding of manifold embeddings.

Contribution

It introduces a new approach involving an exact sequence and explicit actions on embeddings, extending classification results for high-dimensional manifolds.

Findings

Derived estimations for isotopy classes of embeddings

Established an exact sequence involving Whitney invariants

Utilized parametric connected sum techniques

Abstract

We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H_{k+1}(N;Z_2) on the set of embeddings. (For k\ne1 classification results were obtained by algebraic methods without direct construction of embeddings or homology invariants.) The proof involves reduction to classification of embeddings of punctured manifold and uses parametric connected sum of embeddings.