Journal of Topology最新文献

The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

Let $��(�\Gamma �,�P�)�$ be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of $��(�\Gamma �,�P�)�$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of $�\Gamma$ and no restriction on the peripheral subgroups $��P�\in �P�$.

We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a $�{\mathrm{spin}}^c$ 4-manifold with boundary and with an involution that reverses the $�{\mathrm{spin}}^c$ structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3-manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov-type inequalities that relate topological quantities of 4-manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4-manifolds, and nonsmoothable unoriented surfaces in 4-manifolds.

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying $�{\tau }_G$, defined by the second author via the minus flavors $�{\underline{KHI}}^{-}$ and $�{\underline{KHM}}^{-}$ of the knot homologies, with $�{\tau }_G^{♯}$, defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute $�{\underline{KHI}}^{-}$ and $�{\underline{KHM}}^{-}$ for twist knots.

Proposition 6.9 in (J. Topol. 15 (2022), no. 4, 2270–2297) is incorrect without a connectivity assumption. In this note, we provide a counter-example, give a correct proof of the modified proposition and explain the other changes that need to be made to [1].

We give an expression for the Smith–Thom deficiency of the Hilbert square $�X^{�\left[�2�\right]�}$ of a smooth real algebraic variety $�X$ in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of $�X^{�\left[�2�\right]�}$ in the case of projective complete intersections, and show that with a few exceptions, no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.

In a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor $�E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $�f$ is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe $�U$. When specialized to an appropriate family, this produces a localization which when interpreted in the $�\infty$-topos of spaces agrees with the localization corresponding to $�E$. Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any $�\infty$-topos. Second, while the local objects produced by Casacuberta et al. are always 1-types, our construction can produce $�n$-types, for any $�n$. This is new, even in the $�\infty$-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.