Journal of Topology最新文献

We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has, in fact, two natural mirrors, one of which is a derived scheme. These two mirrors are related via a nongeometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.

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This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in étale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number of results about the étale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber–Gabber affine analogue of the proper basechange theorem, and Fujiwara–Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt–Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite étale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of Künneth formulas for the étale homotopy type.

We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinzón-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns that act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative $�\tau$-invariant. Our results also recover and extend several theorems in this area established using instanton Floer homology.

The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.

In this paper, we study the left-orderability of 3-manifold groups using an enhancement, called recalibration, of Calegari and Dunfield's ‘flipping’ construction, used for modifying $��{\text{Homeo}}_{+}��\left(�S^1�\right)��$-representations of the fundamental groups of closed 3-manifolds. The added flexibility accorded by recalibration allows us to produce $��{\text{Homeo}}_{+}��\left(�S^1�\right)��$-representations of hyperbolic link exteriors so that a chosen element in the peripheral subgroup is sent to any given rational rotation. We apply these representations to show that the branched covers of families of links associated to epimorphisms of the link group onto a finite cyclic group are left-orderable. This applies, for instance, to fibred hyperbolic strongly quasi-positive links. Our result on the orderability of branched covers implies that the degeneracy locus of any pseudo-Anosov flow on an alternating knot complement must be meridional, which generalises the known result that the fractional Dehn twist coefficient of any hyperbolic fibred alternating knot is zero. Applications of these representations to order detection of slopes are also discussed in the paper.

By considering a particular type of invariant Seifert surfaces we define a homomorphism $�\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots $�\stackrel{\sim }{C}$ to a new equivariant algebraic concordance group $�{\stackrel{\sim }{G}}^Z$. We prove that $�\Phi$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism (J. Lond. Math. Soc. (2023), no. 107, 2025-2053) and Alfieri and Boyle's equivariant signature (Michigan Math. J. 1 (2023), no. 1, 1–17). Moreover, we provide a partial result on the isomorphism type of $�{\stackrel{\sim }{G}}^Z$ and obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox–Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $�\Phi$ can obstruct equivariant sliceness for knots with Alexander polynomial one.

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain $��6�k�$-dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of $��6�k�$-dimensional manifolds with a metric of positive Ricci curvature.