Journal of Topology最新文献

For a compact subset $�K$ of a closed symplectic manifold $��(�M�,�\omega �)�$, we prove that $�K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.

We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well-behaved theories of power operations for $�E_{\infty }$ ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction-theoretic machinery, we obtain tools for computing with $�E_{\infty }$ algebras over $�F_p$ and over Lubin–Tate spectra. As an application, we demonstrate the existence of $�E_{\infty }$ periodic complex orientations at heights $��h�⩽�2�$.

We prove that for all $��k�,�m�,�n�\in �N�\cup �\left\{�\infty �\right\}�$ with $��4�⩽�k�⩽�m�⩽�n�$, there exists a finitely generated group $�G$ with a finitely generated subgroup $�H$ such that $��asdim�\left(�G�\right)�=�k�$, $��{asdim}_{�A�N�}��\left(�G�\right)��=�m�$, and $��{asdim}_{�A�N�}��\left(�H�\right)��=�n�$. This simultaneously answers two open questions in asymptotic dimension theory.

We consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let $��k�\left(�n�\right)�$ be the smallest positive integer such that any integral $�n$-dimensional homology class becomes realizable in the sense of Steenrod after multiplication by $��k�\left(�n�\right)�$. The best known upper bound for $��k�\left(�n�\right)�$ was obtained independently by Brumfiel and Buchstaber in 1969. All known lower bounds for $��k�\left(�n�\right)�$ were very far from this upper bound. The main result of this paper is a new lower bound for $��k�\left(�n�\right)�$ that is asymptotically equivalent to the Brumfiel–Buchstaber upper bound (in the logarithmic scale). For , we prove that our lower bound is exact. Also, we obtain analogous results for the case of realization of integral homology classes by continuous images of smooth stably complex manifolds.

In this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf.