Right-angled Coxeter quandles and polyhedral products

Kishimoto, Daisuke

doi:10.1090/proc/14534

Right-angled Coxeter quandles and polyhedral products
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by Daisuke Kishimoto PDF

Proc. Amer. Math. Soc. 147 (2019), 3715-3727 Request permission

Abstract:

To a Coxeter group $W$ one associates a quandle $X_W$ from which one constructs a group $\mathrm {Ad}(X_W)$. This group turns out to be an intermediate object between $W$ and the associated Artin group. By using a result of Akita, we prove that $\mathrm {Ad}(X_W)$ is given by a pullback involving $W$, and by using this pullback, we show that the classifying space of $\mathrm {Ad}(X_W)$ is given by a space called a polyhedral product whenever $W$ is right-angled. Two applications of this description of the classifying space are given.

References

T. Akita, The adjoint group of a Coxeter quandle, Kyoto J. Math. (to appear).
N. Andruskiewitsch, F. Fantino, G. A. García, and L. Vendramin, On Nichols algebras associated to simple racks, Groups, algebras and applications, Contemp. Math., vol. 537, Amer. Math. Soc., Providence, RI, 2011, pp. 31–56. MR 2799090, DOI 10.1090/conm/537/10565
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), no. 3, 1634–1668. MR 2673742, DOI 10.1016/j.aim.2010.03.026
Noel Brady, Jonathan P. McCammond, Bernhard Mühlherr, and Walter D. Neumann, Rigidity of Coxeter groups and Artin groups, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 91–109. MR 1950875, DOI 10.1023/A:1020948811381
Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. MR 1104531, DOI 10.1215/S0012-7094-91-06217-4
Michael W. Davis and Boris Okun, Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products, Groups Geom. Dyn. 6 (2012), no. 3, 485–531. MR 2961283, DOI 10.4171/GGD/164
Graham Denham and Alexander I. Suciu, Moment-angle complexes, monomial ideals and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson., 25–60. MR 2330154, DOI 10.4310/PAMQ.2007.v3.n1.a2
Michael Eisermann, Quandle coverings and their Galois correspondence, Fund. Math. 225 (2014), no. 1, 103–168. MR 3205568, DOI 10.4064/fm225-1-7
Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. MR 1392221, DOI 10.1007/BFb0094429
Kouyemon Iriye and Daisuke Kishimoto, Decompositions of suspensions of spaces involving polyhedral products, Algebr. Geom. Topol. 16 (2016), no. 2, 825–841. MR 3493409, DOI 10.2140/agt.2016.16.825
Kouyemon Iriye and Daisuke Kishimoto, Fat-wedge filtration and decomposition of polyhedral products, Kyoto J. Math. 59 (2019), no. 1, 1–51. MR 3934622, DOI 10.1215/21562261-2017-0038
Takefumi Nosaka, Central extensions of groups and adjoint groups of quandles, Geometry and analysis of discrete groups and hyperbolic spaces, RIMS Kôkyûroku Bessatsu, B66, Res. Inst. Math. Sci. (RIMS), Kyoto, 2017, pp. 167–184. MR 3821082
T. E. Panov and Ya. A. Verëvkin, Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups, Mat. Sb. 207 (2016), no. 11, 105–126 (Russian, with Russian summary); English transl., Sb. Math. 207 (2016), no. 11-12, 1582–1600. MR 3588981, DOI 10.4213/sm8701
Mentor Stafa, On the fundamental group of certain polyhedral products, J. Pure Appl. Algebra 219 (2015), no. 6, 2279–2299. MR 3299732, DOI 10.1016/j.jpaa.2014.09.001
J. H. C. Whitehead, On the asphericity of regions in a $3$-sphere, Fund. Math. 32 (1939), no. 1, 149–166.