Quasi-projective K(π,1) conjecture for Artin groups

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Let Γ\Gamma be a labeled graph such that its associated Artin group AΓ\mathbb{A}_\Gamma is quasi-projective. A smooth, connected, quasi-projective Eilenberg–MacLane space is a space whose higher homotopy groups vanish and whose fundamental group is the specified group.

Quasi-projective K(π,1)K(π,1) conjecture. There exists a smooth, connected, quasi-projective Eilenberg–MacLane space XX such that

AΓ=π1(X).\mathbb{A}_\Gamma=\pi_1(X).

This asks whether every quasi-projective Artin group admits a quasi-projective aspherical realization, strengthening the existence of a quasi-projective realization of its fundamental group. The supplied text gives no evidence that the assertion has been resolved.

References

Primary source

Ruben Blasco-Garcia and Jose I. Cogolludo-Agustin, “Quasi-projectivity of even Artin groups”, arXiv:1803.05274 (2018).

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