Topology and homotopy of lattice isomorphic arrangements

TL;DR

This paper demonstrates that arrangements with lattice isomorphism can have topologically distinct embeddings in the complex projective plane, despite having equivalent fundamental groups or homotopy types, and provides explicit examples.

Contribution

It establishes the existence of lattice isomorphic arrangements with homotopy-equivalent complements but non-homeomorphic embeddings, including explicit real and complex examples.

Findings

Existence of arrangements with lattice isomorphism and non-homeomorphic embeddings

Examples of arrangements with homotopy-equivalent complements but different embeddings

Explicit real and complex arrangements illustrating the phenomenon

Abstract

We prove the existence of lattice isomorphic line arrangements having $\pi_1$-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is formed by real-complexified arrangements while the second is not.