Journal of Topology最新文献

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.

In this paper, we prove the equivalence of two symmetric monoidal $�\infty$-categories of $�\infty$-operads, the one defined in Lurie [Higher algebra, available at the author's homepage, http://math.ias.edu/~lurie/, September 2017 version] and the one based on dendroidal spaces.

Symplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux. We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand–Cetlin fibres of Fano varieties in terms of their polytopes. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.

We construct a spectral sequence converging to the Lubin–Tate theory, that is, Morava $�E$-theory, of unordered configuration spaces and identify its $�E^2$-page as the homology of a Chevalley–Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $�E$-theory of the weight $�p$ summands of iterated loop spaces of spheres (parameterizing the weight $�p$ operations on $�E_n$-algebras), as well as the $�E$-theory of the configuration spaces of $�p$ points on a punctured surface. We read off the corresponding Morava $�K$-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the $�F_p$-homology of the space of unordered configurations of $�p$ particles on a punctured surface.

Given an equivariant knot $�K$ of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on $�K$. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of $�S^3$ and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of $�K$ for several equivariant knots.

This paper determines the full, derived deformation theory of certain smooth rational curves $�C$ in Calabi–Yau 3-folds, by determining all higher $�A_{\infty }$-products in its controlling DG-algebra. This geometric setup includes very general cases where $�C$ does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3-fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the non-commutative deformation theory of $�C$ is described via a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari (Adv. Theor. Math. Phys. 7 (2003) 619–665), Aspinwall–Katz (Comm. Math. Phys.. 264 (2006) 227–253) and Curto–Morrison (J. Algebraic Geom. 22 (2013) 599–627). Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3-folds, which lift Artin's (Amer. J. Math. 84 (1962) 485–496) celebrated results from surfaces.

In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain $�M$ of dimension $��2�n�$ is $��(�n�-�1�)�$-Calabi–Yau, assuming that the wrapped Fukaya category of $�M$ admits an at most countable set of Lagrangians that generate it and satisfy some finiteness condition on morphism spaces between them.