Topology and homotopy of lattice isomorphic arrangements

Guerville-Ballé, Benoît

doi:10.1090/proc/14878

Topology and homotopy of lattice isomorphic arrangements
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by Benoît Guerville-Ballé PDF

Proc. Amer. Math. Soc. 148 (2020), 2193-2200 Request permission

Abstract:

We prove the existence of lattice isomorphic line arrangements having $\pi _1$-equivalent or homotopy-equivalent complements and non-homeomorphic embeddings in the complex projective plane. We also provide two explicit examples: one is formed by real-complexified arrangements, while the second is not.

References

Enrique Artal Bartolo, Jorge Carmona Ruber, José Ignacio Cogolludo-Agustín, and Miguel Marco Buzunáriz, Topology and combinatorics of real line arrangements, Compos. Math. 141 (2005), no. 6, 1578–1588. MR 2188450, DOI 10.1112/S0010437X05001405
Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín, Benoît Guerville-Ballé, and Miguel Marco-Buzunáriz, An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111 (2017), no. 2, 377–402. MR 3623846, DOI 10.1007/s13398-016-0298-y
Enrique Artal Bartolo, Benoît Guerville-Ballé, and Juan Viu-Sos, Fundamental groups of real arrangements and torsion in the lower central series quotients, Experimental Mathematics, 2018, DOI: 10.1080/10586458.2018.1428131
Pi. Cassou-Noguès, C. Eyral, and M. Oka, Topology of septics with the set of singularities $\mathbf {B}_{4,4}\oplus 2\mathbf {A}_3\oplus 5\mathbf {A}_1$ and $\pi _1$-equivalent weak Zariski pairs, Topology Appl. 159 (2012), no. 10-11, 2592–2608. MR 2923428, DOI 10.1016/j.topol.2012.04.011
Graham Denham and Alexander I. Suciu, Torsion in the homology of Milnor fibers of hyperplane arrangements, Combinatorial methods in topology and algebra, Springer INdAM Ser., vol. 12, Springer, Cham, 2015, pp. 31–36. MR 3467321, DOI 10.1007/978-3-319-20155-9_{7}
Christophe Eyral and Mutsuo Oka, $\pi _1$-equivalent weak Zariski pairs, Tokyo J. Math. 28 (2005), no. 2, 499–526. MR 2191063, DOI 10.3836/tjm/1244208204
Michael Falk, Homotopy types of line arrangements, Invent. Math. 111 (1993), no. 1, 139–150. MR 1193601, DOI 10.1007/BF01231283
Michael J. Falk and Nicholas J. Proudfoot, Parallel connections and bundles of arrangements, Topology Appl. 118 (2002), no. 1-2, 65–83. Arrangements in Boston: a Conference on Hyperplane Arrangements (1999). MR 1877716, DOI 10.1016/S0166-8641(01)00042-6
Michael Falk and Richard Randell, On the homotopy theory of arrangements. II, Arrangements—Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 93–125. MR 1796895, DOI 10.2969/aspm/02710093
Kwai-Man Fan, Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J. 44 (1997), no. 2, 283–291. MR 1460414, DOI 10.1307/mmj/1029005704
Benoît Guerville-Ballé, An arithmetic Zariski 4-tuple of twelve lines, Geom. Topol. 20 (2016), no. 1, 537–553. MR 3470721, DOI 10.2140/gt.2016.20.537
Benoît Guerville-Ballé and Juan Viu-Sos, Configurations of points and topology of real line arrangements, Math. Ann. 374 (2019), no. 1-2, 1–35. MR 3961303, DOI 10.1007/s00208-018-1673-0
Helmut A. Hamm and Lê Dũng Tráng, Lefschetz theorems on quasiprojective varieties, Bull. Soc. Math. France 113 (1985), no. 2, 123–142 (English, with French summary). MR 820315, DOI 10.24033/bsmf.2023
Tan Jiang and Stephen S.-T. Yau, Topological invariance of intersection lattices of arrangements in $\textbf {C}\textrm {P}^2$, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 88–93. MR 1197426, DOI 10.1090/S0273-0979-1993-00409-9
Mutsuo Oka and Koichi Sakamoto, Product theorem of the fundamental group of a reducible curve, J. Math. Soc. Japan 30 (1978), no. 4, 599–602. MR 513072, DOI 10.2969/jmsj/03040599
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. MR 558866, DOI 10.1007/BF01392549
G. L. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, Funktsional. Anal. i Prilozhen. 45 (2011), no. 2, 71–85 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 45 (2011), no. 2, 137–148. MR 2848779, DOI 10.1007/s10688-011-0015-8
Ichiro Shimada, Equisingular families of plane curves with many connected components, Vietnam J. Math. 31 (2003), no. 2, 193–205. MR 2011641
Taketo Shirane, A note on splitting numbers for Galois covers and $\pi _1$-equivalent Zariski $k$-plets, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1009–1017. MR 3589301, DOI 10.1090/proc/13298
Kristopher Williams, On the homotopy type of the complement of an arrangement that is a 2-generic section of the parallel connection of an arrangement and a pencil of lines, Available at arXiv:1507.04706, 2015.