First-hand information for everyone
These notes explain a proof, aimed at non-specialists, of a long-standing geometric statement about Artin groups in the spherical case. Roughly speaking, the statement—called the K(π,1) conjecture—says that if you remove a symmetric finite collection of flat walls (hyperplanes) from complex space C^n, the remaining space is a classifying space for a naturally associated group. A classifying space, also written K(G,1), is a space whose only nontrivial homotopy group is the fundamental group G. The paper focuses on the case where the symmetry group is finite; this is called the spherical case.
The notes start by recalling the basic objects: Coxeter groups and their associated Artin groups. A familiar example is the symmetric group on n letters and its Artin group, the braid group on n strands. The geometric object of interest is the orbit configuration space Y_W: a quotient of C^n with the hyperplanes removed and then divided by the Coxeter group action. A standard fact used here is that the fundamental group of Y_W is the Artin group G_W. The K(π,1) conjecture asks whether Y_W is a classifying space for G_W.
To prove the conjecture in the spherical case the author uses combinatorial topology tools. A central object is the Salvetti complex, a finite cell complex that models the topology of Y_W. Its cells are indexed by certain finite subgroups (called spherical standard parabolic subgroups). The paper explains how to apply discrete Morse theory, a combinatorial method for simplifying cell complexes while preserving their essential topology, to the Salvetti complex. In effect, the complex can be simplified enough to show it has the homotopy type required for a K(G,1) space in the spherical setting.
Why this matters: showing Y_W is a classifying space gives a concrete topological model for the Artin group G_W. That model lets mathematicians compute algebraic invariants that come from topology, such as group cohomology, and it clarifies the shape of spaces related to braids and reflection groups. The spherical case already includes important examples like braid groups and free abelian groups, so the result covers many classical situations.
Important caveats are noted in the text. The proof presented is for spherical type only, meaning it requires the Coxeter group to be finite. The spherical case is a classical result: the original proof in this setting goes back to Pierre Deligne (1972). These notes give a different, more combinatorial proof that follows the same core idea. Other cases of the conjecture are known in the literature (for example certain affine and low-dimensional cases), but the full conjecture remains broader than what is treated here.
Context and provenance: the notes are based on a mini-course given at WinterBraids 2023 and draw on survey work by Paris and on the author’s master’s thesis. The excerpt available is a set of lecture-style notes that develop the Coxeter and Artin group background, introduce the Salvetti complex explicitly, and then use discrete Morse theory to reach the spherical K(π,1) conclusion.