Homotopy type through homology groups

TL;DR

This paper establishes conditions under which a topological complex's homology groups determine it to have a homotopy type equivalent to a wedge of spheres, extending known results to other dimension pairs.

Contribution

It generalizes the relationship between homology groups and homotopy types for complexes with specific homology conditions across multiple dimension pairs.

Findings

Complexes with certain homology conditions are homotopy equivalent to wedges of spheres.

Generalizations to other pairs of dimensions are provided.

Results connect algebraic homology data with topological homotopy types.

Abstract

We show that if a complex has free finitely generated reduced homology groups for two consecutive dimensions and trivial homology for all other dimensions, then it must have the homotopy type of a wedge of spheres of two consecutive dimensions. We also show other pairs of dimensions for which the last result can be generalized.