#### Volume 15, issue 5 (2015)

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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
##### Abstract

Let $f:{ℂ}^{n+1}\to ℂ$ be a polynomial that is transversal (or regular) at infinity. Let $\mathsc{U}={ℂ}^{n+1}\setminus {f}^{-1}\left(0\right)$ 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 $\mathsc{U}$ 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 $\mathsc{U}$.

##### Keywords
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
##### Publication
Received: 13 June 2014
Accepted: 13 February 2015
Published: 10 December 2015
##### Authors
 Yongqiang Liu School of Mathematical Sciences University of Science and Technology of China No 96, JinZhai Road Hefei, 230026 China http://home.ustc.edu.cn/~liuyq/ Laurenţiu Maxim Department of Mathematics University of Wisconsin–Madison 480 Lincoln Drive Office 713 Madison, WI 53706-1388 USA http://www.math.wisc.edu/~maxim/