Journal of Topology最新文献

Let $�\omega$ be a Kähler form on the real 4-torus $�T^4$. Suppose that $�\omega$ satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of $�\omega$. This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of $��(�T^4�,�\omega �)�$ is infinitely generated.

Given a lattice Veech group in the mapping class group of a closed surface $�S$, this paper investigates the geometry of $�\Gamma$, the associated $��{\pi }_1�S�$-extension group. We prove that $�\Gamma$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of $�\Gamma$ on a hyperbolic space, retaining most of the geometry of $�\Gamma$. This action is a key ingredient in the sequel where we show that $�\Gamma$ is hierarchically hyperbolic and quasi-isometrically rigid.

We provide a sufficient condition for splitness of a link in terms of its reduced $��\mathrm{sl}�\left(�N�\right)�$ link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If $�N$ is prime and the coefficient field is of characteristic $�N$, then the sufficient condition for splitness is also necessary. When $��N�=�2�$, we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with $��Z�/�2�$ coefficients.

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander polynomials. We also give formulae relating the taut polynomial and the untwisted Alexander polynomial. There are two formulae, depending on whether the maximal free abelian cover of a veering triangulation is edge-orientable or not. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case, we give formulae that relate the specialisation of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmüller polynomial and the Alexander polynomial to the non-fibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball.

We give an obstruction to positive scalar curvature metrics on 4-manifolds with the homology $��S^1�\times �S^3�$ described in terms of homology cobordism invariants from Seiberg–Witten theory. The main tool of the proof is a relative Bauer–Furuta-type invariant on a periodic-end 4-manifold.

We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3-manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3-manifolds with totally geodesic boundary.

We show that the sheaf of $�A^1$-connected components of a reductive algebraic group over a perfect field is strongly $�A^1$-invariant. As a consequence, torsors under such groups give rise to $�A^1$-fiber sequences. We also show that sections of $�A^1$-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $�R$-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.