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Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Yongqiang Liu and Laurenţiu Maxim

Algebraic & Geometric Topology 15 (2015) 2755–2785

Let f : n+1 be a polynomial that is transversal (or regular) at infinity. Let U = n+1 f1(0) be the corresponding affine hypersurface complement. By using the peripheral complex associated to f, we give several estimates for the (infinite cyclic) Alexander polynomials of U induced by f, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of U.

Reidemeister torsion, Sabbah specialization complex, nearby cycles, peripheral complex, hypersurface complement, Milnor fibre, non-isolated singularities, Alexander polynomial, boundary manifold, mixed Hodge structure
Mathematical Subject Classification 2010
Primary: 32S25
Secondary: 32S55, 32S60
Received: 13 June 2014
Accepted: 13 February 2015
Published: 10 December 2015
Yongqiang Liu
School of Mathematical Sciences
University of Science and Technology of China
No 96, JinZhai Road
Hefei, 230026
Laurenţiu Maxim
Department of Mathematics
University of Wisconsin–Madison
480 Lincoln Drive
Office 713
Madison, WI 53706-1388