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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 \(H^{\ell+1}(M,{\mathbb C})=0.\)

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
Full Text: DOI arXiv
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