Journal of Topology最新文献

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.

This paper gives three different proofs (independently obtained by the three authors) of the following fact: given an Anosov flow on an oriented 3-manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $�R$-covered positively twisted.

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces.

Given a finitely generated group $�G$, the possible actions of $�G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $��(�Y�,�\Phi �)�$ naturally associated with $�G$ and uniquely defined up to flow equivalence, that we call the Deroin space of $�G$. We show a realisation result: every expansive flow $��(�Y�,�\Phi �)�$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions.

To a complex polynomial function $�f$ with arbitrary singularities, we associate the number of Morse points in a general linear Morsification $��f_t�:�=�f�-�t�\ell �$. We produce computable algebraic formulae in terms of invariants of $�f$ for the numbers of stratwise Morse trajectories that abut, as $��t�\to �0�$, to some point of the space or at infinity.