×

zbMATH — the first resource for mathematics

Generic section of a hyperplane arrangement and twisted Hurewicz maps. (English) Zbl 1159.52023
Let \(\mathcal A\) be an essential arrangement of affine hyperplanes in \({\mathbb C}^\ell,\) with complement \(M={\mathbb C}^\ell \setminus \bigcup {\mathcal A}.\) While \(M\) is in general not an aspherical space, it is known that the Hurewicz homomorphism \(\pi_n(M,x_0) \to H_n(M,{\mathbb Z})\) is trivial for all \(n\geq 2.\) This is a consequence of the fact that \(H^*(M, {\mathbb Z})\) is free abelian and generated by degree-one classes [see R. Randell, Topol. Appl. 78, 201–213 (1997; Zbl 0880.55007)].
If \(\mathcal L\) is a local system of \({\mathbb Z}\)-modules over \(M,\) then there is a twisted version of the Hurewicz map. If \(n\geq 2\) and \(f:S^n \to M,\) then \(f^*{\mathcal L}\) is trivial, and the assignment \(f \otimes t \mapsto f_*([S^n]\otimes t)\) determines a homomorphism \(h:\pi_n(M,x_0)\otimes {\mathcal L}_{x_0} \to H_n(M,{\mathcal L}).\) In the paper under review, the author shows that, if \(M\) is a generic section of a hyperplane complement \(M'\) in \({\mathbb C}^{\ell+1},\) and \({\mathcal L}\) is the restriction of a local system \({\mathcal L}'\) on \(M'\) satisfying some specific non-resonance conditions, then the twisted Hurewicz homomorphism \(h:\pi_\ell(M,x_0)\otimes {\mathcal L}_{x_0} \to H_\ell(M,{\mathcal L})\) is surjective, hence nontrivial.
The proof relies on the fact \(M\) has the homotopy type of an \(\ell\)-dimensional CW complex, and, up to homotopy type, \(M'\) is obtained from \(M\) by attaching the minimal number \(b_{\ell+1}(M')\) of \((\ell+1)\)-cells. The attaching maps are then homologically trivial. Since \(b_\ell(M)=b_\ell(M')\) it follows that the images under \(h\) of the attaching maps generate \(H_\ell(M,{\mathcal L}).\)
As a corollary of the main result, one deduces that \(M\) is not aspherical, but, as the author notes, there is an elementary proof of this fact: \(H^*(\pi_1(M,x_0),{\mathbb C})\) must map onto \(H^*(M',{\mathbb C}),\) since \(\pi_1(M,x_0)\cong \pi_1(M',x_0)\) and \(H^*(M',{\mathbb C})\) is generated in degree one, but \(H^{\ell+1}(M',{\mathbb C})\neq 0\) while \(H^{\ell+1}(M,{\mathbb C})=0.\)

MSC:
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
32S99 Complex singularities
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P99 Homotopy theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Cho, K., A generalization of kita and Noumi’s vanishing theorems of cohomology groups of local system, Nagoya math. J., 147, 63-69, (1997) · Zbl 0912.32023
[2] Cohen, D.C.; Dimca, A.; Orlik, P., Nonresonance conditions for arrangements, Ann. inst. Fourier (Grenoble), 53, 6, 1883-1896, (2003) · Zbl 1054.32016
[3] Dimca, A.; Papadima, S., Hypersurface complements Milnor fibers and minimality of arrangements, Ann. of math., 158, 473-507, (2003) · Zbl 1068.32019
[4] Dimca, A.; Papadima, S., Equivariant chain complexes, twisted homology and relative minimality of arrangements, Ann. sci. √©cole norm. sup. (4), 37, 3, 449-467, (2004) · Zbl 1059.32007
[5] Hamm, H., Lefschetz theorems for singular varieties, singularities, Proc. sympos. pure math., vol. 40, (1983), pp. 547-557
[6] 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, 2, 205-219, (1975) · Zbl 0306.55011
[7] Kohno, T., Homology of a local system on the complement of hyperplanes, Proc. Japan acad. ser. A, 62, 144-147, (1986) · Zbl 0611.55005
[8] Orlik, P.; Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. math., 56, 167-189, (1980) · Zbl 0432.14016
[9] Orlik, P.; Terao, H., Arrangements of hyperplanes, (1992), Springer-Verlag · Zbl 0757.55001
[10] Orlik, P.; Terao, H., Arrangements and hypergeometric integrals, MSJ mem., 9, (2001) · Zbl 0980.32010
[11] Papadima, S.; Suciu, A., Higher homotopy groups of complements of complex hyperplane arrangements, Adv. math., 165, 1, 71-100, (2002) · Zbl 1019.52016
[12] Randell, R., Homotopy and group cohomology of arrangements, Topology appl., 78, 201-213, (1997) · Zbl 0880.55007
[13] Randell, R., Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. amer. math. soc., 130, 2737-2743, (2002) · Zbl 1004.32010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.