Twisted homology jump loci, twisted Alexander polynomials, and $Σ$-invariants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917540582653952 |
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| author | Liu, Yongqiang Suciu, Alexander I. |
| author_facet | Liu, Yongqiang Suciu, Alexander I. |
| contents | The twisted Alexander polynomials of a space, associated to a linear representation $σ$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants -- the twisted homology jump loci -- which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri--Neumann--Strebel--Renz (BNSR) $Σ$-invariants.
For compact orientable $3$-manifolds with toroidal or empty boundary, we use a theorem of Friedl--Vidussi to show that the closure of the union of these twisted tropical bound is sharp: it recovers the fibered faces of the Thurston norm ball exactly, a result that fails without twisting. For compact Kähler manifolds, we prove that the $Σ^1$-invariant of $π_1(X)$ is controlled by the orbifold fibrations of $X$ for any representation $σ$, and that the twisted Alexander polynomial $Δ^σ(X)$ must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twisted homology jump loci, twisted Alexander polynomials, and $Σ$-invariants Liu, Yongqiang Suciu, Alexander I. Algebraic Geometry Group Theory Geometric Topology 20J05, 57M07, 14T05, 20F65, 32Q15, 55N25, 57M05, 57M27 The twisted Alexander polynomials of a space, associated to a linear representation $σ$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants -- the twisted homology jump loci -- which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri--Neumann--Strebel--Renz (BNSR) $Σ$-invariants. For compact orientable $3$-manifolds with toroidal or empty boundary, we use a theorem of Friedl--Vidussi to show that the closure of the union of these twisted tropical bound is sharp: it recovers the fibered faces of the Thurston norm ball exactly, a result that fails without twisting. For compact Kähler manifolds, we prove that the $Σ^1$-invariant of $π_1(X)$ is controlled by the orbifold fibrations of $X$ for any representation $σ$, and that the twisted Alexander polynomial $Δ^σ(X)$ must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts. |
| title | Twisted homology jump loci, twisted Alexander polynomials, and $Σ$-invariants |
| topic | Algebraic Geometry Group Theory Geometric Topology 20J05, 57M07, 14T05, 20F65, 32Q15, 55N25, 57M05, 57M27 |
| url | https://arxiv.org/abs/2605.28595 |