We construct non-Kähler Calabi–Yau manifolds of dimension ⩾ 4 with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties. The examples have K3 fibrations over smooth projective varieties and their algebraic dimensions are of codimension 2.

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of  G and  K is invertible in a given principal ideal domain  k. It is known that in this case the cohomology of the homogeneous space  G / K with coefficients in  k and the torsion product of  H ∗ ( B K ) and  k over  H ∗ ( B G ) are isomorphic as k-modules

We show that the Morse–Novikov number of knots in S 3 is additive under connected sum and unchanged by cabling.

Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group C ̂ Z (respectively, C Z ) was previously defined as the set of knots in homology spheres that bounds homology balls (respectively, in S 3 ), modulo homology concordance. We prove C ̂ Z / C Z contains a Z ∞ subgroup. We construct our family of examples by applying the filtered mapping cone formula to

We compare two naturally arising notions of ‘unknotting number’ for 2-spheres in the 4-sphere: namely, the minimal number of 1-handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2-sphere. We refer to these invariants as the stabilization number and the Casson–Whitney number of the sphere, respectively

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper, we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end, we compute the structure of the tmf-resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E 1 -page of the tmf-resolution

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all

We define two concordance invariants of knots using framed instanton homology. These invariants ν ♯ and τ ♯ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the τ invariant in Heegaard Floer homology. We use ν ♯ and τ ♯ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z\colon \mathcal{B}\to \hat{A}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\hat{A}$ is the degree-completion of the category $\mathbf{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of $\mathfrak{sl}_2$. From the above we

Arzhantseva proved that every infinite-index quasi-convex subgroup H of a torsion-free hyperbolic group G is a free factor in a larger quasi-convex subgroup of G. We give a probabilistic generalization of this result. That is, we show that when R is a subgroup generated by independent random walks in G, then ⟨ H , R ⟩ ≅ H * R with probability going to one as the lengths of the random walks go to infinity

We show that the graph TQFT for Heegaard Floer homology satisfies a strong version of Atiyah's duality axiom for a TQFT. As an application, we compute some Heegaard Floer mixed invariants of 4-dimensional mapping tori in terms of Lefschetz numbers on HF + .

We prove a surgery formula of the Casson–Seiberg–Witten invariant of integral homology S 1 × S 3 along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg–Witten invariants or a manifestation of the surgery exact triangle in four-dimensional Seiberg–Witten theory of homology S 1 × S 3 . As an application, we compute this invariant for mapping tori of

We prove that the automorphism group of every infinitely ended finitely generated group is acylindrically hyperbolic. In particular Aut ( F n ) is acylindrically hyperbolic for every n ⩾ 2 . More generally, if G is a group which is not virtually cyclic, and hyperbolic relative to a finite collection P of finitely generated proper subgroups, then Aut ( G , P ) is acylindrically hyperbolic.

We describe a procedure to deform cubulations of hyperbolic groups by ‘bending hyperplanes’. Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic 3-manifolds and free-by- Z groups, and Hsu–Wise turns. As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits a proper, cocompact, essential action on a

Let H < PSL 2 ( Z ) be a finite index normal subgroup which is contained in a principal congruence subgroup, and let Φ ( H ) ≠ H denote a term of the lower central series or the derived series of H . In this paper, we prove that the commensurator of Φ ( H ) in PSL 2 ( R ) is discrete. We thus obtain a natural family of thin subgroups of PSL 2 ( R ) whose commensurators are discrete, establishing some

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce basic moves that allow the transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This allows us to show that the target of this spectral sequence can be identified with Hochschild homology when the topological Hochschild homology is torsion-free as a module over THH ∗ ( F 2 ) , parallel

We prove that every symplectic 4‐manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in CP 2 can be symplectically isotoped into bridge position. This paper relies extensively on

In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the Pin(2)-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we explore further this non-formality phenomenon. Specifically, we provide explicit descriptions of several Massey products induced

We define a faithful linear monoidal functor from the partition category, and hence from Deligne’s category $\mathrm{\underline{Re}p}(S_t)$, to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.

For any positive integer $k$ and $p\in \{3,5,7\}$ we construct a link which has a direct summand $\mathbb Z/p^k\mathbb Z$ in its Khovanov cohomology.

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper

We prove a concordance version of the 4‐dimensional light bulb theorem for π 1 ‐negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if F 0 and F 1 are such surfaces in a 4‐manifold X that are homotopic and there exists an immersed framed 2‐sphere G in X intersecting F 0 geometrically once, then F 0 and F 1 are concordant if

We find conditions under which a non‐orientable closed surface smoothly embedded into an orientable 4 ‐manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4 ‐sphere. This allows us to extend the Gabai 4 ‐dimensional light bulb theorem and the Auckly–Kim–Melvin–Ruberman–Schwartz ‘one is enough’ theorem to the case of non‐orientable surfaces

We show that non‐elliptic prime 3‐manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

For a punctured surface S , we characterize the representations of its fundamental group into PSL 2 ( C ) that arise as the monodromy of a meromorphic projective structure on S with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo–Kapovich–Marden concerning C P 1 ‐structures on closed surfaces, and settles a long‐standing question about characterizing

We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2 ℵ 0 quasi‐isometry classes, provided that every non‐empty open subset of S contains at least two non‐quasi‐isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2 ℵ 0 quasi‐isometry classes. These results account for most known constructions

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

The complement of an arrangement of hyperplanes in C n has a natural bordification to a manifold with corners formed by removing (or “blowing up”) tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces

We generalize the Mahowald invariant to the R ‐motivic and C 2 ‐equivariant settings. For all i > 0 with i ≡ 2 , 3 mod 4 , we show that the R ‐motivic Mahowald invariant of ( 2 + ρ η ) i ∈ π 0 , 0 R ( S 0 , 0 ) contains a lift of a certain element in Adams' classical v 1 ‐periodic families, and for all i > 0 , we show that the R ‐motivic Mahowald invariant of η i ∈ π i , i R ( S 0 , 0 ) contains a

The physical 3d $\mathcal N = 2$ theory $T[Y]$ was previously used to predict the existence of some $3$-manifold invariants $\widehat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten–Reshetikhin–Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author in [4]. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern–Simons perturbation theory. The $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding $\mathfrak {gl}_2$ skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of $\mathfrak {gl}_2$ foams that admits an interesting

Let M be a smooth projective variety and D an ample normal crossings divisor. From topological data associated to the pair ( M , D ) , we construct, under assumptions on Gromov–Witten invariants, a series of distinguished classes in symplectic cohomology of the complement X = M ∖ D . Under further ‘topological’ assumptions on the pair, these classes can be organized into a log(arithmic) PSS morphism

We compute mod ( p , v 1 ) topological Hochschild homology of the connective cover of the K ( 1 ) ‐local sphere spectrum for all primes p ⩾ 3 . This is accomplished using a May‐type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.

For a finite group G , we define an equivariant cobordism category C d G . Objects of the category are ( d − 1 ) ‐dimensional closed smooth G ‐manifolds and morphisms are smooth d ‐dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (that is, geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of a certain equivariant

We show that quadratic and symmetric L ‐theory of the integers are related by Anderson duality and that both spectra split integrally into the L ‐theory of the real numbers and a generalised Eilenberg–Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G / Top . Finally, we prove analogous results for the genuine L‐spectra recently devised for the study of Grothendieck–Witt theory

In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_\infty$-algebras.

The homotopy category of the bordism category h C d has as objects closed oriented ( d − 1 ) ‐manifolds and as morphisms diffeomorphism classes of d ‐dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of h C d for d = 1 , exhibiting it as a circle bundle over Ω ∞ − 2 CP − 1 ∞ .

We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent

Suppose G is a 1‐ended finitely generated group that is hyperbolic relative to P, a finite collection of 1‐ended finitely generated proper subgroups of G . Our main theorem states that if the boundary ∂ ( G , P ) has no cut point, then G has semistable fundamental group at ∞ . Under mild conditions on G and the members of P, the 1‐ended hypotheses and the no cut point condition can be eliminated to

Let ( S 2 , ω ) be a symplectic sphere, and let τ : S 2 → S 2 be an anti‐symplectic involution of ( S 2 , ω ) . We consider the product ( S 2 , ω ) × ( S 2 , ω ) endowed with the anti‐symplectic involution τ × τ , and study the space of monotone anti‐invariant symplectic forms on this four‐manifold. We show that this space is disconnected. In addition, during the course of the proof, we produce a diffeomorphism

Kronheimer and Mrowka introduced a new knot invariant, called s ♯ , which is a gauge theoretic analogue of Rasmussen's s invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi‐positive knots with non‐trivial right‐handed torus knots. These computations reveal some unexpected phenomena: we show that

We prove that the group of homeomorphisms of the circle introduced by the author with Justin Moore is of type F ∞ . This makes the group the first example of a type F ∞ group which is nonamenable and does not contain nonabelian free subgroups. To prove our result, we provide a certain generalisation of cube complexes, which we refer to as cluster complexes. We also obtain a computable normal form,

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of B U . This presents an E ∞ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E ∞ ‐ring M U P . Here, we prove that these two E ∞ ‐rings are in fact different, though the underlying E 2 ‐rings are equivalent. Nonetheless, we prove

Given a connected cobordism between two knots in the 3‐sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band‐unlinking number of a knot, the fusion number of a ribbon knot

Let X be a nonsingular projective algebraic variety over C , and let M ¯ g , n , β ( X ) be the moduli space of stable maps f : ( C , x 1 , … , x n ) → X from genus g , n ‐pointed curves C to X of degree β . Let S be a line bundle on X . Let A = ( a 1 , ⋯ , a n ) be a vector of integers which satisfy ∑ i = 1 n a i = ∫ β c 1 ( S ) . Consider the following condition: the line bundle f ∗ S has a meromorphic

For an ample line bundle L on a complete toric surface X , we consider the subset V L ⊂ | L | of irreducible, nodal, rational curves contained in the smooth locus of X . We study the monodromy map from the fundamental group of V L to the permutation group on the set of nodes of a reference curve C ∈ V L . We identify a certain obstruction map Ψ X defined on the set of nodes of C and show that the image

For a suitable choice of the cube category, we construct a Grothendieck topology on it such that sheaves with respect to this topology are exactly simplicial sets (thus establishing simplicial sets as a reflective subcategory of cubical sets). We then extend the construction of the homotopy coherent nerve to cubical categories and establish an analogue of Lurie's straightening–unstraightening construction

We prove the unrolled superalgebra $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $\xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.

Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(\Sigma)$ and its associated graded ${\rm gr}\, \mathfrak{g}(\Sigma)$

We give a purely combinatorial formula for evaluating closed, decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral, equivariant version of colored Khovanov–Rozansky link homology categorifying the $\mathfrak {sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag varieties.

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one‐to‐one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z / 2 Z , we define A - Tangles to be the category consisting of A ‐tangles, which are balanced tangles

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$

This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics

We prove a generalization of the classical connectivity theorem of Blakers–Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system ( L , R ) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chachólski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016)

In this paper we complete a chain of explicit Quillen equivalences between the model category for Θ n + 1 ‐spaces and the model category of small categories enriched in Θ n ‐spaces. The Quillen equivalences given here connect Segal category objects in Θ n ‐spaces, complete Segal objects in Θ n ‐spaces, and Θ n + 1 ‐spaces.