On Levin's generalization of the plus construction

TL;DR

This paper proves Levin's theorem that any connected CW complex can be extended to a simply connected one by attaching 2- and 3-dimensional cells, preserving homology in higher dimensions.

Contribution

It provides a proof of Levin's generalization of the plus construction, showing how to construct simply connected complexes with preserved higher homology.

Findings

Construction of a simply connected CW complex from any connected CW complex.

Preservation of homology groups in dimensions greater than one.

Extension of Levin's theorem to a broader class of complexes.

Abstract

We present a proof of the following theorem of Levin: For every connected CW complex $K$ there is a simply connected CW complex $K^+$ obtained from $K$ by attaching cells of dimension 2 and 3 such that the inclusion $K\to K^+$ induces isomorphisms of homology groups in dimension $>1$.