On the homotopy classification of spaces by the fixed loop space homology

TL;DR

This paper develops a classification method for certain R-local CW-complexes up to homotopy using their loop space homology, extending the understanding of homotopy types through algebraic invariants.

Contribution

It introduces a complete homotopy invariant based on loop space homology for R-local CW-complexes with specific connectivity and dimension constraints.

Findings

Classifies R-local spaces up to homotopy using loop space homology.

Constructs a complete invariant for spaces with given loop space homology.

Provides a framework for homotopy classification in algebraic topology.

Abstract

Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty $) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1, $ a complete homotopy invariant is constructed in terms of the loop space homology $H_*(\Omega X).$ This allows us to classify all such $R$-local spaces up to homotopy with a fixed loop space homology.