Polyhedral fans are geometric objects, which arise naturally in many areas of mathematics, for example in toric geometry, the theory of hyperplane arrangements and representation theory. In many cases, there are natural ways of identifying some of the polyhedral cones defining a fan, thus giving a “partition of the fan”. To each such partitioned fan we associate the category of a partitioned fan. The classifying spaces of these categories turn out to be cube complexes whenever the fan is simplicial, and hence we are interested in studying their topological properties.

On the other hand, there is often a natural poset on the maximal-dimensional cones of a fan. From this poset and the partition, we can construct a group we call the picture group. Our guiding question is whether the classifying spaces of the categories are K(pi,1) spaces, in other words have non-trivial homotopy group only in degree 1, for this picture group. We give an example of a partitioned hyperplane arrangement and prove that the classifying space is a K(pi,1) space for the picture group obtained from the poset of regions.

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