Abstract
I present a result according to which the complement of any affine 2-arrangement in \(\mathbb{R}^{d}\) is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its ith Betti number. To this end, we prove that the Björner–Ziegler complement complexes, induced by combinatorial stratifications of any essential 2-arrangement, admit perfect discrete Morse functions. This result extend previous work by Falk, Dimca–Papadima, Hattori, Randell, and Salvetti–Settepanella, among others.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adiprasito, K.: Combinatorial stratifications and minimality of 2-arrangements (2012, preprint). arXiv:1211.1224. http://arxiv.org/abs/1211.1224
Adiprasito, K., Benedetti, B.: Subdivisions, shellability and the Zeeman conjecture (2012, preprint). arXiv:1202.6606. http://arxiv.org/abs/1202.6606
Artal Bartolo, E., Carmona Ruber, J., Cogolludo Agustín, J.I., Marco Buzunáriz, M.Á.: Invariants of combinatorial line arrangements and Rybnikov’s example. In: Singularity Theory and Its Applications. Advanced Studies in Pure Mathematics, vol. 43, pp. 1–34. Mathematical Society of Japan, Tokyo (2006)
Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. In: Lectures on Modern Mathematics, vol. II, pp. 93–128. Wiley, New York (1964)
Björner, A., Ziegler, G.M.: Combinatorial stratification of complex arrangements. J. Am. Math. Soc. 5, 105–149 (1992)
Cohen, D.C., Suciu, A.I.: Homology of iterated semidirect products of free groups. J. Pure Appl. Algebra 126, 87–120 (1998). doi:10.1016/S0022-4049(96)00153-3. http://dx.doi.org/10.1016/S0022-4049(96)00153-3
Dimca, A., Papadima, S.: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements. Ann. Math. 158(2), 473–507 (2003). doi:10.4007/annals2003.158.473. http://dx.doi.org/10.4007/annals.2003.158.473
Falk, M.J.: Homotopy types of line arrangements. Invent. Math. 111, 139–150 (1993). doi:10.1007/BF01231283. http://dx.doi.org/10.1007/BF01231283
Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)
Goresky, R.M., MacPherson, R.D.: Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14. Springer, Berlin (1988)
Hamm, H., Lê, D.T.: Un théorème du type de Lefschetz. C. R. Acad. Sci. Paris Sér. A-B 272, A946–A949 (1971)
Hattori, A.: Topology of C n minus a finite number of affine hyperplanes in general position. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22, 205–219 (1975)
Kirby, R.C.: Problems in low–dimensional topology (1995). Available online at Robion Kirby’s webpage, http://math.berkeley.edu/~kirby/problems.ps.gz
Papadima, S., Suciu, A.: The spectral sequence of an equivariant chain complex and homology with local coefficients. Trans. Am. Math. Soc. 362(5), 2685–2721 (2010)
Randell, R.: Morse theory, Milnor fibers and minimality of hyperplane arrangements. Proc. Am. Math. Soc. 130, 2737–2743 (2002). doi:10.1090/S0002-9939-02-06412-2. http://dx.doi.org/10.1090/S0002-9939-02-06412-2
Rybnikov, G.L.: On the fundamental group of the complement of a complex hyperplane arrangement. Technical Report, DIMACS, Rutgers University, Piscataway (1994)
Rybnikov, G.L.: On the fundamental group of the complement of a complex hyperplane arrangement. Funktsional. Anal. i Prilozhen. 45, 71–85 (2011)
Salvetti, M.: Topology of the complement of real hyperplanes in C n. Invent. Math. 88, 603–618 (1987)
Salvetti, M., Settepanella, S.: Combinatorial Morse theory and minimality of hyperplane arrangements. Geom. Topol. 11, 1733–1766 (2007). doi:10.2140/gt.2007.11.1733. http://dx.doi.org/10.2140/gt.2007.11.1733
Yoshinaga, M.: Hyperplane arrangements and Lefschetz’s hyperplane section theorem. Kodai Math. J. 30, 157–194 (2007). doi:10.2996/kmj/1183475510. http://dx.doi.org/10.2996/kmj/1183475510
Zeeman, E.C.: On the dunce hat. Topology 2, 341–358 (1964)
Ziegler, G.M.: On the difference between real and complex arrangements. Math. Z. 212, 1–11 (1993). doi:10.1007/BF02571638. http://dx.doi.org/10.1007/BF02571638
Ziegler, G.M., Živaljević, R.T.: Homotopy types of arrangements via diagrams of spaces. Math. Ann. 295, 527–548 (1993)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Adiprasito, K.A. (2015). Combinatorial Stratifications and Minimality of Two-Arrangements. In: Benedetti, B., Delucchi, E., Moci, L. (eds) Combinatorial Methods in Topology and Algebra. Springer INdAM Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-20155-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-20155-9_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20154-2
Online ISBN: 978-3-319-20155-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)