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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
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##### 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. école 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
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