Journal of Topology最新文献

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof,, we obtain a motivic version of mod $�k$ Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety $���\left(�N_{{\mathrm{GL}}_r}�T�\right)���\setminus ��{\mathrm{GL}}_r�$ which turns out to be not stably $�A^1$-connected. We also show that the higher motivic stable stems are of bounded torsion.

We develop a general theory of pushforward operations for principal $�G$-bundles equipped with a certain type of orientation. In the case $��G�=��B�U�\left(�1�\right)��$ and orientations in twisted K-theory, we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation. As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible $��B�U�\left(�1�\right)�$-action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite solvable quotient of the fundamental group we exhibit a cover whose homology is not generated by the lifts of curves in the complement of its kernel. We explain how the existing approach of Malestein and Putman (for branched covers) relates to the Magnus embedding, and by doing so we simplify their construction. We then generalise it to unbranched covers by producing embeddings of surface groups into units of certain graded associative algebras, which may be of independent interest.

The proof of [2, Lemma 5.1] is incomplete as it relies on some results in [4], the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of [2, Lemma 5.1].

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.