Journal of Topology最新文献

We develop a mechanism of “isotropy separation for compact objects” that explicitly describes an invertible $�G$-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of possible combinations of geometric fixed points of invertible $�G$-spectra in the case $��G�=�A_5�$. A further application is given by showing that the Picard groups of $�{Sp}^G$ and a category of derived Mackey functors agree.

The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four-ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an “infinite staircase,” that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair-step pattern. Later work of Usher and Cristofaro-Gardiner–Holm–Mandini–Pires suggested that while four-dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly nongeneric. This paper concludes the systematic study of one-point blowups of the complex projective plane, building on previous work of Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff, Magill-McDuff-Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro-Gardiner–Holm–Mandini–Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be $��1�/�3�$. We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres).

We give a self-contained and streamlined rendition of Andrea Bianchi's recent proof of the Mumford conjecture using moduli spaces of branched covers.

Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer $�k$, new hyperbolic groups admitting surjective homomorphisms to $�Z$ and to $�Z^2$, whose kernel is of type $�F_k$ but not of type $�F_{�k�+�1�}$. By a fibre product construction, we also find examples of non-normal subgroups of Kähler groups with exotic finiteness properties.

We describe a new algebro-geometric perspective on the study of the Milnor fibration and, as a first step toward putting it into practice, prove powerful criteria for a deformation of a holomorphic function germ to admit a stratification on its domain partially satisfying the Thom condition and, more generally, to respect the Milnor fibration of the original germ in an appropriate sense. As corollaries, we obtain a method of partitioning the space of homogeneous polynomials of a fixed degree into finitely many locally closed subsets such that the fiber diffeomorphism type of the Milnor fibration is constant along each subset and a criterion under which deformations of a function with critical locus a complete intersection will be well-behaved.

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.

We characterize a certain neck-pinching degeneration of (marked) $��C�P^1�$-structures on a closed oriented surface $�S$ of genus at least two. In a more general setting, we take a path of $��C�P^1�$-structures $��C_t���(�t�⩾�0�)��$ on $�S$ that leaves every compact subset in its deformation space, such that the holonomy of $�C_t$ converges in the $��{\mathrm{PSL}}_2�C�$-character variety as $��t�\to �\infty �$. Then, it is well known that the complex structure