Are two finite $H$-spaces homotopy equivalent?

TL;DR

This paper introduces an algorithm to determine whether two simply connected finite simplicial spaces are homotopy equivalent, including their Postnikov stages, and extends to stable homotopy equivalence.

Contribution

It presents the first algorithmic solution for homotopy equivalence of finite simplicial sets with finite $k$-type and dimension, including stable cases.

Findings

Algorithm successfully decides homotopy equivalence.

Algorithm finds explicit homotopy equivalences when they exist.

Extends to stable homotopy equivalence detection.

Abstract

This paper proposes an algorithm that decides if two simply connected spaces represented by finite simplicial sets of finite $k$-type and finite dimension $d$ are homotopy equivalent. If the spaces are homotopy equivalent, the algorithm finds a homotopy equivalence between their Postnikov stages in dimension $d$. As a consequence, we get an algorithm deciding if two spaces represented by finite simplicial sets are stably homotopy equivalent.