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Given a real hyperplane arrangement, the Salvetti complex of the associated oriented matroid is homotopy equivalent to the complement of the complexification of the arrangement. In particular, its cohomology is isomorphic to the Orlik-Solomon algebra of the underlying (unoriented) matroid.

Given an oriented matroid, it is still possible to define both the Salvetti complex and the Orlik-Solomon algebra, and it is a folk theorem that the cohomology of the Salvetti complex is isomorphic to the Orlik-Solomon algebra. This is asserted without proof by Gelfand and Rybnikov in their paper Algebraic and topological invariants of oriented matroids, which is referenced in the book Oriented Matroids (page 95).

Does anyone know if a proof of this statement can be found in the literature?

For complex hyperplane arrangements, the proof that the cohomology of the complement is isomorphic to the Orlik-Solomon algebra involves a deletion/contraction induction that makes use of the Thom isomorphism. It's not immediately clear to me how to make a similar argument with complements replaced by Salvetti complexes.

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  • $\begingroup$ The Thom isomorphism can be avoided because the bundle in question is trivial. By the LES of a triple, you can replace it with the pair (disk bundle, sphere bundle). The Thom isomorphism becomes the usual degree-shifting isomorphism for a reduced suspension. Maybe the Salvetti complex has some combinatorial version of this pair? $\endgroup$
    – John Wiltshire-Gordon
    Commented Feb 19 at 20:45
  • $\begingroup$ Yeah, I wondered about that. With the complements we have $M' = M \sqcup M''$, and I wonder if there is an analogous relationship between the Salvetti complexes. It can't be exactly the same, since the Salvetti complexes are compact. $\endgroup$
    – Nicholas Proudfoot
    Commented Feb 19 at 21:45

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I think, the reference you are looking for is the paper by Björner and Ziegler "Combinatorial Stratification of Complex Arrangements" in JAMS. They give a complete combinatorial proof in Sec.7, using the homology of the link complex and Alexander Duality. The most general result is Theorem 8.5 (formulated in terms of 2-pseudoarrangements).

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  • $\begingroup$ Thanks for this reference! I didn't know about it, and I agree that there appears to be a proof in there. I am going to hold off on accepting it as a final answer in case there are any more direct proofs out there. (For example, it would be nice to not have to realize the oriented matroid by a pseudosphere arrangement.) That said, I really appreciate the answer. $\endgroup$
    – Nicholas Proudfoot
    Commented Feb 20 at 22:15

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