Twisted Hurewicz detection for the coned arrangement cX2

Determine whether the twisted Hurewicz maps from pi_k(M(cX2)) tensor L_x0 to H_k(M(cX2), L) with rank-one local system coefficients L detect the non-triviality of pi_k(M(cX2)), where X2 is the affine line arrangement in the plane consisting of six lines whose complement has fundamental group generated by gamma_1, ..., gamma_6 with relations [gamma_2, gamma_3] = 1, [gamma_4, gamma_5] = 1, [gamma_2, gamma_5] = 1, [gamma_1, gamma_6] = 1, [gamma_3, gamma_6] = 1, [gamma_1, gamma_4] = 1, gamma_1 gamma_3 gamma_5 = gamma_3 gamma_5 gamma_1 = gamma_5 gamma_1 gamma_3, and gamma_2 gamma_4 gamma_6 = gamma_4 gamma_6 gamma_2 = gamma_6 gamma_2 gamma_4; cX2 denotes the coning of X2, and M(cX2) its complexified complement.

Background

The paper develops a method to construct embedded spheres in the complexified complement of a real hyperplane arrangement and to detect their non-triviality via twisted intersection numbers and twisted Hurewicz maps. Section 5 explores obstructions to the K(pi,1) property using locally consistent but globally inconsistent systems of half-spaces.

Example 5.5 presents a specific six-line affine arrangement X2 whose complement is not K(pi,1). Letting A = cX2 be the coning of this arrangement yields a central arrangement whose complement M(cX2) is also not K(pi,1). Despite satisfying a natural combinatorial condition (Sigma_2 = Sigma_3), this example shows that such conditions do not imply the K(pi,1) property.

Motivated by the twisted Hurewicz machinery developed earlier in the paper, the authors ask whether, for this concrete non-K(pi,1) example (the coned arrangement cX2), the non-trivial higher homotopy groups can be detected via twisted Hurewicz maps.