We compute some R$\mathbb {R}$-motivic stable homotopy groups. For s−w⩽11$s - w \leqslant 11$, we describe the motivic stable homotopy groups πs,w$\pi _{s,w}$ of a completion of the R$\mathbb {R}$-motivic sphere spectrum. We apply the ρ$\rho$-Bockstein spectral sequence to obtain R$\mathbb {R}$-motivic Ext$\operatorname{Ext}$ groups from the C$\mathbb {C}$-motivic Ext$\operatorname{Ext}$ groups, which

Bivariant theory is a unified framework for cohomology and Borel–Moore homology theories. In this paper, we extract an ∞$\infty$-enhanced bivariant homology theory from Gaitsgory–Rozenblyum's six functor formalism.

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new strongly invertible concordance group invariants, Donaldson's theorem, and the g-signature. We find many new examples where the equivariant 4-genus is larger than the 4-genus.

For any oriented cusped hyperbolic 3-manifold M $M$ , we study its ( R , ε ) $(R,\epsilon )$ -panted cobordism group, which is the abelian group generated by ( R , ε ) $(R,\epsilon )$ -good curves in M $M$ modulo the oriented boundaries of ( R , ε ) $(R,\epsilon )$ -good pants. In particular, we prove that for sufficiently small ε > 0 $\epsilon >0$ and sufficiently large R > 0 $R>0$ , some modified

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘ S 1 $S^1$ S1 -equivariant

Let M = W ∪ T V $M=\mathcal {W}\cup _\mathcal {T} \mathcal {V}$ be an amalgamation of two compact 3-manifolds along a torus, where W $\mathcal {W}$ is the exterior of a knot in a homology sphere. Let N $N$ be the manifold obtained by replacing W $\mathcal {W}$ with a solid torus such that the boundary of a Seifert surface in W $\mathcal {W}$ is a meridian of the solid torus. This means that there is

We give a short and elementary proof of the nonrealizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.

We introduce a global equivariant refinement of algebraic K-theory; here ‘global equivariant’ refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global Ω $\Omega$ -spectrum that keeps track of genuine G $G$ -equivariant infinite loop spaces, for all finite groups G $G$ . The resulting global algebraic K-theory

For a half-translation surface ( S , q ) $(S,q)$ (S,q) , the associated saddle connection complex A ( S , q ) $\mathcal {A}(S,q)$ A(S,q) is the simplicial complex where vertices are the saddle connections on  ( S , q ) $(S,q)$ (S,q) , with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension n $n$ n from families of such hypersurfaces in ambient dimensions k = 1 , … , n − 1 $k=1,\ldots ,n-1$ k=1,…,n−1 . The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of

We prove that Morse local-to-global groups grow exponentially faster than their infinite-index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds. To accomplish this, we develop a theory of

We explore the cohomological structure for the (possibly singular) moduli of SLnSLn$\mathrm{SL}_n$ -Higgs bundles for arbitrary degree on a genus gg$g$ curve with respect to an effective divisor of degree >2g−2>2g−2$>2g-2$ . We prove a support theorem for the SLnSLn$\mathrm{SL}_n$ -Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel–Thaddeus

Let S $S$ be a surface of negative Euler characteristic and consider a finite filling collection Γ $\Gamma$ of closed curves on S $S$ in minimal position. An observation of Foulon and Hasselblatt shows that P T ( S ) ∖ Γ ̂ $PT(S) \setminus \widehat {\Gamma }$ is a finite-volume hyperbolic 3-manifold, where P T ( S ) $PT(S)$ is the projectivized tangent bundle and Γ ̂ $\widehat \Gamma$ is the set of

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological

In this paper, we study the (in)sensitivity of the Khovanov functor to 4-dimensional linking of surfaces. We prove that if L $L$ L and L ′ $L^{\prime }$ are split links, and C $C$ is a cobordism between L $L$ and L ′ $L^{\prime }$ that is the union of disjoint (but possibly linked) cobordisms between the components of L $L$ and the components of L ′ $L^{\prime }$ , then the map on Khovanov homology

We give a short and independent proof of a theorem of Danciger and Zhang: surface groups with Hitchin linear part cannot act properly on the affine space. The proof is fundamentally different and relies on ergodic methods.

This work focuses on important step in quantitative topology: given homotopic mappings from S m $S^m$ to S n $S^n$ of Lipschitz constant L $L$ , build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant plays a role: m = 3 $m = 3$ , n = 2 $n = 2$ , constructing a homotopy with

We prove an ‘h-principle without pre-conditions’ for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that such tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner { p = | q | } ⊂ R 2 $\lbrace p=|q|\rbrace \subset

Hopkins and Mahowald gave a simple description of the mod p $p$ Eilenberg Mac Lane spectrum F p ${\mathbb {F}}_p$ as the free E 2 $\mathbb {E}_2$ -algebra with an equivalence of p $p$ and 0. We show for each faithful 2-dimensional representation λ $\lambda$ of a p $p$ -group G $G$ that the G $G$ -equivariant Eilenberg Mac Lane spectrum F ̲ p $\underline{\mathbb {F}}_p$ is the free E λ $\mathbb {E}_{\lambda

We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (that is, non-classical) twist of K $K$ -theory over G = S U ( n ) $G=SU(n)$ . This twist is represented by a Fell bundle E → G $\mathcal {E}\rightarrow \mathcal {G}$ , which reduces to the basic gerbe for the top exterior power functor. The

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer–Witt K $K$ -theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence, we lift the fundamental fiber sequence of η $\eta$ -periodic motivic stable homotopy theory established in Bachmann and Hopkins (2020) from fields to arbitrary

Let X $X$ X be a compact complex space in Fujiki's Class C $\mathcal {C}$ C . We show that the group Aut ( X ) $\operatorname{Aut}(X)$ Aut(X) of all biholomorphic automorphisms of X $X$ X has the Jordan property: there is a (Jordan) constant J = J ( X ) $J = J(X)$ J=J(X) such that any finite subgroup G ⩽ Aut ( X ) $G\leqslant \operatorname{Aut}(X)$ has an abelian subgroup H ⩽ G $H\leqslant G$ with

We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant- ∞ $\infty$ -operads in

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

Let 𝐺G$G$ be an affine algebraic group defined over a field 𝑘k$k$ of characteristic 0. We study the derived moduli space of 𝐺G$G$ -local systems on a pointed connected CW complex 𝑋X$X$ trivialized at the basepoint of 𝑋X$X$ . This derived moduli space is represented by an affine DG scheme RLoc𝐺(𝑋,*)RLocG(X,*)$ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call the (co)homology of the structure

We introduce a notion of freeness for R O $RO$ 𝑅𝑂 -graded equivariant generalized homology theories, considering spaces or spectra E $E$ 𝐸 such that the R $R$ 𝑅 -homology of E $E$ 𝐸 splits as a wedge of the R $R$ 𝑅 -homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation

In [Doc. Math. 18 (2013), 1045–1087], Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof proceeds by first recubulating to obtain an improper action with controlled

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step

We fix an error in the proof of Theorem 6.1(ii) in Watanabe (J. Topol. 2 (2009), no. 3, 624–660), and extend the main result of that paper to all odd dimensions at least 5.

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values in the space of oriented geodesics of H n + 1 $\mathbb {H}^{n+1}$ , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion G $G$ of the universal cover of an n $n$ -manifold M $M$ , equivariant for some group representation

In the previous work, the author used symplectic flexibility techniques to prove an upper bound on the number of generators of the wrapped Fukaya category of a high-dimensional, simply connected Weinstein domain. In this article, we give a purely categorical proof of this result for all Weinstein domains via Thomason's theorem on split-generating subcategories and the Grothendieck group. In the process

We construct an invariant of closed spin c $\mathrm{spin}^c$ 4-manifolds using families of Seiberg–Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing

We consider a complex smooth projective variety equipped with an action of an algebraic torus with a finite number of fixed points. We compare the motivic Chern classes of Białynicki-Birula cells with the 𝐾K$K$ -theoretic stable envelopes of a cotangent bundle. We prove that under certain geometric assumptions for example for homogenous spaces, these two notions coincide up to normalization.

We define and parametrize so-called sl ( 2 ) $\mathfrak {sl}(2)$ -type fibres of the Sp ( 2 n , C ) $\mathsf {Sp}(2n,\mathbb {C})$ - and SO ( 2 n + 1 , C ) $\mathsf {SO}(2n+1,\mathbb {C})$ -Hitchin system. These are (singular) Hitchin fibres, such that spectral curve establishes a 2-sheeted covering of a second Riemann surface Y $Y$ . This identifies the sl ( 2 ) $\mathfrak {sl}(2)$ -type Hitchin fibres

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a 33$\hskip.001pt 3$ -manifold or a null-homologous knot inside a 33$\hskip.001pt 3$ -manifold and the Heegaard diagram of that 33$\hskip.001pt 3$ -manifold or knot

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

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

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