Abstract For each integer we construct an oriented closed simply connected 4‐manifold admitting a smoothly embedded closed connected surface of self‐intersection number such that the complement of the surface has non‐trivial fundamental group. This answers a question of Kronheimer in Kirby's 1997 problem list. The proof combines a topological construction with homological properties of simple groups such as Thompson's group and certain sporadic finite simple groups.
Let and be knot manifolds and be the closed manifold obtained by gluing up and via . We show that if admits a co-oriented taut foliation, then identifies some CTF-detected rational boundary slopes of and , affirming a conjecture proposed in Boyer, McA Gordon, and Hu [Adv. Math. 495 (2026), 110956].
We define a new notion of splitting complexity for a group along a non-trivial integral character . If is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism equals the -Euler characteristic of the kernel of . This allows us to define a Thurston-type semi-norm which measures the splitting complexity of integral characters. Our main tool is Friedl-L & uuml;ck's -polytope which we compute for all groups in this class. We also compute the -Betti numbers of all kernels of characters and show that the -homology of any finitely generated subgroup of is concentrated in the first dimension, which may be of independent interest.
We calculate the ordinary C-2-cohomology of BT2 with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to that of BT1 and BU(2), calculated previously, both relationships being more complicated than in the nonequivariant case.
If is a fibered knot in a closed, oriented 3-manifold with fiber , and has rank , then the monodromy of is freely isotopic to a diffeomorphism with at most fixed points. This generalizes earlier work of Baldwin-Hu-Sivek and Ni. We also clarify a misleading formula in Cotton-Clay's computation of the symplectic Floer homology of mapping classes of surfaces.
We prove explicit linear stable ranges for the -modules and with being the configuration -space of a -dimensional manifold with . The proof of this result uses a homotopy-theoretic approach to representation stability for -modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to -modules and orbit configuration spaces.
Let be a compact symplectic manifold with convex boundary and . Suppose that is equipped with a convex Hamiltonian -action for some connected, compact Lie group . We construct an action of the pure Coulomb branch of on the -equivariant symplectic cohomology of . Building on work of Teleman (J. Eur. Math. Soc. 23 (2021), 3497-3520), we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima (Adv. Theor. Math. Phys. 22 (2018), 1071-1147) in terms of equivariant symplectic cohomology.
Let be a finite group and be an -ring -spectrum. For any -space and positive integer , we give an explicit description of the smallest Mackey ideal in for which the reduced th power operation is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of -Green functors. This theorem also specializes to characterize the appropriate ideal when is a -ring in global spectra. We give example computations for the sphere spectrum, complex -theory, and Morava -theory.
The original Arnold chord conjecture states that every closed Legendrian submanifold of the standard contact sphere admits a Reeb chord with distinct endpoints with respect to any contact form. In this paper, we prove this conjecture for contact forms induced by strictly convex embeddings into under the assumption that minimal periodic Reeb orbits are of Morse-Bott type. We also provide a counterexample when the convexity condition is not satisfied.
We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3-manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one-ended virtually torsion-free graphically discrete groups are action rigid within the class of virtually torsion-free groups. We also prove quasi-isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi-convex subgroups. This includes the case of two hyperbolic 3-manifold groups amalgamated along a quasi-convex malnormal non-abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.
We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that Pic(E_n/I) = ℤ/2, where E_n is a Lubin–Tate theory and I is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of K(n)-local generalized Moore algebras, and make some preliminary computations including the height 1 case.
We study finite abelian group actions on weakly Lefschetz cohomologically symplectic (WLS) manifolds, a collection of manifolds that includes all compact connected Kaehler manifolds. We prove that for any WLS manifold X there exists a number C such that, for any integer m≥ C, if (𝐙/m)^k acts freely on X, then ∑_j b_j(X;𝐐)≥ 2^k. We also prove a structure theorem for effective actions on WLS manifolds of (𝐙/p)^r, where p is a big enough prime, analogous to some results for tori of Lupton and Oprea, and we find bounds on the discrete degree of symmetry of WLS manifolds. Our technique, which may be of independent interest, is based on studying the cohomology of abelian covers of WLS manifolds X associated to certain maps π:X→ T^k. We prove that, in the presence of actions of arbitrarily big finite abelian groups, some of these abelian covers have finitely generated cohomology, and the spectral sequence associated to π degenerates at the second page over the rationals.
We describe some periodic structure in the cohomology of the moduli stack of one-dimensional formal group laws, also known as the -page of the classical Adams-Novikov spectral sequence. This structure is distinct from the familiar -periodicities, and it displays interesting number-theoretic properties. Our techniques involve the -motivic Adams spectral sequence, and we obtain analogous periodic structure in -motivic stable homotopy.
We prove the existence of surface subgroups within any cocompact lattice Γ in SO(2n,1) for n≥2. This result addresses the cases missing from the work of Hamenstädt in 2015, who constructed surface subgroups in cocompact lattices for all other rank-one semisimple Lie groups of non-compact type.
We introduce the concept of a pants decomposition for a finitely generated free group and construct the corresponding pants graph. A pants decomposition of a free group leads to the formation of a simplicial graph, referred to as the pants graph of a free group, consisting of all possible pants decompositions.The natural isometric action of the outer automorphism group of the free group on the pants graph induces a coarsely surjective orbit map.Additionally, we construct a coarsely Lipschitz map from the pants graph to the free splitting complex. These results imply that the pants graph of a free group is both connected and unbounded.
We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.
We define a new notion of splitting complexity for a group G along a non-trivial integral character ϕ∈ H^1(G; ℤ). If G is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism ϕ G →ℤ equals the L^2-Euler characteristic of the kernel of ϕ. This allows us to define a Thurston-type semi-norm ·_T H^1(G ; ℝ) →ℝ that measures the splitting complexity of integral characters. Our main tool is Friedl–Lück's L^2-polytope.
Surgery on a knot in is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of alternating surgery slopes is algorithmically computable and to establish several structural results. Furthermore, we calculate the set of alternating surgery slopes for many examples of knots, including all hyperbolic knots in the SnapPy census. These examples exhibit several interesting phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes. We also establish upper bounds on the set of alternating surgeries, showing that an alternating surgery slope on a hyperbolic knot satisfies . Notably, this bound applies to lens space surgeries, thereby strengthening the known genus bounds from the conjecture of Goda and Teragaito.
