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Homotopy groups and twisted homology of arrangements. (English) Zbl 1173.55003
M. Yoshinaga [Topology Appl. 155, No. 9, 1022–1026 (2008; Zbl 1159.52023)] showed that the twisted Hurewicz homomorphism \(h: \pi_k(M'') \otimes_{ \mathbb Z} {\mathcal L} \to H_k(M'',{\mathcal L})\) is surjective, if \(M''\) is a generic hyperplane section of the complement \(M\) of an arrangement of hyperplanes in \({\mathbb C}^{k+1},\) and \({\mathcal L}\) is a generic local system on \(M''.\) (In this context, \({\mathcal L}\) is generic if and only if \(H_{k-1}(M'',{\mathcal L})=0.\)) The argument involves some special features of arrangement complements, in particular, that they are minimal spaces, meaning they have CW decompositions for which the attaching maps are homologically trivial.
In the paper under review, Randell shows that surjectivity of \(h\) is a simple consequence of the twisted relative Hurewicz theorem, the naturality of \(h\), and the fact that the pair \((M,M'')\) is \((k-1)\)-connected. In particular, minimality of \(M\) is not necessary. The general result is: if \((X,Y)\) is a \((k-1)\)-connected topological pair with \(k\geq 3,\) and \(H_{k-1}(Y,N)=0,\) then \(h\) is surjective. Here \(N\) is a \({\mathbb Z}[\pi]\)-module, \(\pi=\pi_1(Y),\) and \(h: \pi_k(Y)\otimes_{{\mathbb Z}[\pi]} N \to H_k(Y,N).\) Yoshinaga’s result is a direct consequence. A similar argument also shows that the well-known fact that \(H_{k-1}(M'')\to H_{k-1}(M)\) is an isomorphism follows from the triviality of the absolute (untwisted) Hurewicz homomorphism, proved by the author in earlier work.
In the last section the author generalizes the result that the absolute Hurewicz map is trivial for arrangement complements, by giving an upper bound on the image in terms of the homology of the associated covering space, in case \(N={\mathbb Z}[\pi/\pi']\) for \(\pi'\) a subgroup of \(\pi.\) The paper closes with several examples, and a general question: which elements of \(\pi_k(M)\) are detected by local system homology?
MSC:
55N25 Homology with local coefficients, equivariant cohomology
57N65 Algebraic topology of manifolds
55Q52 Homotopy groups of special spaces
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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References:
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