We investigate when finite-order Hamiltonian diffeomorphisms extend to Hamiltonian circle actions, probing the transition from discrete to continuous symmetry in symplectic topology. Focusing on irrational ruled symplectic $4$-manifolds, we show that homologically trivial symplectic cyclic actions of order $k>2$ always extend to Hamiltonian $S^1$-actions, possibly after modifying the symplectic form. In contrast, we construct explicit symplectic involutions that cannot be so extended, even on minimal irrational ruled surfaces. These examples reveal geometric obstructions to extending discrete symmetries and highlight new exotic symplectic actions not equivalent to holomorphic ones. Our results also apply to higher-dimensional and noncyclic group actions, and we establish several structural results on the isomorphism types of finite groups that can act on irrational ruled symplectic $4$-manifolds.

In the homology cobordism group $\Theta_\mathbb{Z}^3$, it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu and Zemke introduced the local equivalence group $\mathfrak{I}$ along with the homomorphism $h\;:\;\Theta_\mathbb{Z}^3 \rightarrow \mathfrak{I}$. Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under h. Therefore, it is interesting to ask if such dependences in $\mathfrak{I}$ originate from $\Theta_\mathbb{Z}^3$. In this paper, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato and Taniguchi, we provide certain conditions to guarantee that such relations are not realised even in the rational homology cobordism group. We also discuss the local equivalence class of the $\operatorname{Pin(2)}$-equivariant Seiberg–Witten Floer stable homotopy type.

We determinewhen an exotic sphere $\Sigma $ of dimension $d\not {\equiv }1\ (\mathrm {mod}\ 4)$ can be detected through the homotopy type of its truncated $\mathscr {D}\mathrm {isc}$-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in $\Sigma $ with natural point-forgetting and -splitting maps between them, and it gives rise to the finite stages in Goodwillie–Weiss’ embedding calculus tower. Our proof involves three ingredients that could be of independent interest: a gluing result for $\mathscr {D}\mathrm {isc}$-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context of $\mathscr {D}\mathrm {isc}$-presheaves, and a computation of the finite residual of the mapping class group of the connected sums $\sharp ^g(S^{2k+1}\times S^{2k+1})$.

We prove that Grothendieck-Witt spaces of Poincaré categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubi’s classical hermitian and quadratic $\mathrm K$-groups with the genuine Grothendieck-Witt groups from our joint work with Calmès, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory.

The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincaré category as the homotopy type of the associated cobordism category. In memory of Bruce Williams.

We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern–Simons functional and prove a localization result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else ‘escape to infinity’. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localizing critical points. As an application, we obtain formulas for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by 4) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold.

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems having a similar behaviour at infinity as the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The principal technical challenge is dealing with the non-compactness of the associated energy hypersurfaces.

In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures.

In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin$^c$ structure determines a family of Dirac operators $\{D_B\}$ parametrized by a $b_1(Y)$-dimensional torus $\mathbb {T}_Y$. In this paper, we develop techniques to study how the topology of the locus $\mathsf {K}\subset \mathbb {T}_Y$ corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus $Y=T^3$ for which the spectral gap $\lambda _1^*$ on coexact $1$-forms is large, after a small perturbation of the family, the locus $\mathsf {K}$ is a two-sphere.

While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak {s})$ with a large $\lambda _1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the topology of the family of Dirac operators $\{D_B\}$.

We show that any finitely presented group with an index two subgroup is realised as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilise after adding a single copy of the complex projective plane.

Let $\mathbb {k}$ be a field, and let $\mathcal {C}$ be a Cauchy complete $\mathbb {k}$-linear braided category with finite-dimensional morphism spaces and . We call an indecomposable object X of $\mathcal C$ non-negligible if there exists $Y\in \mathcal {C}$ such that is a direct summand of $Y\otimes X$. We prove that every non-negligible object $X\in \mathcal {C}$ such that $\dim \operatorname {End}(X^{\otimes n})<n!$ for some n is automatically rigid. In particular, if $\mathcal {C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal {C}$ is rigid. As applications, we simplify Huang’s proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal {C}$, the data of a $\mathcal {C}$-modular functor is equivalent to a modular fusion category structure on $\mathcal {C}$, answering a question of Bakalov and Kirillov. Furthermore, we show that if $\mathcal {C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal {C}$ is zero. Hence $\mathcal {C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.

Kreck proved that two 2q-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q{-}1)$-smoothings over the same normal $(q{-}1)$-type $(B,\xi)$. We show that ‘stably diffeomorphic’ can be replaced by ‘diffeomorphic’ if the normal smoothings have isomorphic Q-forms (consisting of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley’s Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k{-}1)$-connected 4k-manifold with the kernel of a certain bordism map. By the calculations of Senger and Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected 4k-manifold M with normal $(2k{-}1)$-type $(B,\xi)$ such that $H_{2k}(B)$ is free, an algebraic description of the stable class of M, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to M. Using this description, we explicitly compute the stable class of manifolds M with rank-2 hyperbolic intersection form.

Let M be a smooth manifold with $\dim M\geq 3$ and a base point $x_{0}$. Surgeries along the oriented circle $S^{1}\times \{x_{0}\}$ on the product $S^{1}\times M$ yield two manifolds $\Sigma _{0}M$ and $\Sigma _{1}M$, called the suspensions of M.

These suspension operations play a fundamental role in the construction and classification of smooth manifolds admitting free circle actions. This paper presents corresponding results and supporting evidence.

A correction of a corollary in the above-mentioned paper ‘Long time behavior of leafwise heat flow for Riemannian foliations’.

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a ${\text{Pin}}^{\pm}$-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of ${\text{Pin}}^{+}$ and ${\text{Pin}}^-$-structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off ${\text{Pin}}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold M admits a ${\text{Pin}}^-$-structure and we find a criterion to check whether or not it admits a ${\text{Pin}}^+$-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of ${\text{Pin}}^+$-structures on vector bundles.

We introduce a framework to prove integral rigidity results for the Seiberg–Witten invariants of a closed $4$-manifold X containing a nonseparating hypersurface Y satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if X has the homology of a four-torus, and it contains a nonseparating three-torus, then the sum of all Seiberg–Witten invariants of X is determined in purely cohomological terms.

Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson’s TQFT approach to the formula of Meng–Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg–Witten equations on X and reducible ones on Y and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline {\mathit {HM}}_*$ induced by a negative-definite cobordism between three-manifolds, which might be of independent interest.

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.

We establish a version of Seiberg–Witten Floer K-theory for knots, as well as a version of Seiberg–Witten Floer K-theory for 3-manifolds with involution. The main theorems are 10/8-type inequalities for knots and for involutions. The 10/8-inequality for knots yields numerous applications to knots, such as lower bounds on stabilizing numbers and relative genera. We also give obstructions to extending involutions on 3-manifolds to 4-manifolds, and detect non-smoothable involutions on 4-manifolds with boundary.

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

Gay and Meier asked if a trisection diagram for the Gluck twist on a spun or twist-spun 2-knot in $S^4$ obtained by a certain method is standard. In this paper, we show that the trisection diagram for the Gluck twist on the spun $(p+1,p)$-torus knot is standard, where p is any integer greater than or equal to 2.