A note on the homotopy type of the Alexander dual

TL;DR

This paper explores the homotopy properties of the Alexander dual of simplicial complexes, revealing that the dual's homotopy type isn't always determined by the original, and establishing conditions under which the dual resembles a sphere.

Contribution

It introduces new conditions for the homotopy type of the Alexander dual to be spherical and extends duality concepts to reduced lattices.

Findings

Homotopy type of K does not determine that of K*

Constructed simply connected complexes with arbitrary fundamental groups

Identified conditions for K* to have the homotopy type of a sphere

Abstract

We investigate the homotopy type of the Alexander dual of a simplicial complex. In general the homotopy type of K does not determine the homotopy type of its dual K*. Moreover, one can construct for each finitely presented group G, a simply connected simplicial complex K with fundamental group isomorphic to G. We study sufficient conditions on K for K* to have the homotopy type of a sphere. We also extend the simplicial Alexander duality to the context of reduced lattices.