Rational homotopy type of subspace arrangements with a geometric lattice

TL;DR

This paper characterizes when the complement of a subspace arrangement with a geometric lattice is rationally elliptic, showing it is a product of odd spheres under specific conditions, and provides a complete classification.

Contribution

It provides a complete characterization of the rational homotopy type of arrangement complements with geometric lattices, linking ellipticity to direct sum conditions.

Findings

Complement is rationally elliptic iff the sum of orthogonal subspaces is a direct sum.

Homotopy type of the complement is a product of odd-dimensional spheres.

Most arrangements have hyperbolic complements, not elliptic.

Abstract

Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and show that most arrangements have an hyperbolic complement.