We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group  μ ℓ $\mu _\ell$ of ℓ $\ell$ th roots of unity, where ℓ $\ell$ is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group  S ℓ $S_\ell$ and to  μ ℓ $\mu _\ell$ , and introduce a delayed limit Adams spectral sequence

Let Λ ± $\Lambda _\pm$ be Legendrian submanifolds in the cosphere bundle T ∗ , ∞ M $T^{*,\infty }M$ . Given a Lagrangian cobordism L $L$ of Legendrians from Λ − $\Lambda _-$ to Λ + $\Lambda _+$ , we construct a functor Φ L * : Sh Λ + c ( M ) → Sh Λ − c ( M ) ⊗ C − * ( Ω * Λ − ) C − * ( Ω * L ) ${\mathrm{\Phi}}_{L}^{\ast}:{{\rm Sh}}_{{\mathrm{\Lambda}}_{+}}^{c}(M)\to {{\rm Sh}}_{{\mathrm{\Lambda}}_

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular

We show that, for ε = 1 4000 $\varepsilon =\frac{1}{4000}$ , any action of a finite cyclic group by ( 1 + ε ) $(1+\varepsilon )$ -bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.

The Torelli group of a genus g $g$ oriented surface Σ g $\Sigma _g$ is the subgroup I g $\mathcal {I}_g$ of the mapping class group Mod ( Σ g ) ${\rm Mod}(\Sigma _g)$ consisting of all mapping classes that act trivially on H 1 ( Σ g , Z ) ${\rm H}_1(\Sigma _g, \mathbb {Z})$ . The quotient group Mod ( Σ g ) / I g ${\rm Mod}(\Sigma _g) / \mathcal {I}_g$ is isomorphic to the symplectic group Sp ( 2 g

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in R P d $\mathbb {R}\mathrm{P}^d$ is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.

Let S $S$ be a compact orientable surface of genus g $g$ with marked points in the interior. Franks–Handel (Proc. Amer. Math. Soc. 141 (2013) 2951–2962) proved that if n < 2 g $n<2g$ then the image of a homomorphism from the mapping class group Mod ( S ) ${\rm Mod}(S)$ of S $S$ to GL ( n , C ) ${\rm GL}(n,{\mathbb {C}})$ is trivial if g ⩾ 3 $g\geqslant 3$ and is finite cyclic if g = 2 $g=2$ . The first

Let ω $\omega$ be a Kähler form on the real 4-torus T 4 $T^4$ . Suppose that ω $\omega$ satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of ω $\omega$ . This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of ( T 4 , ω ) $(T^4,\omega )$ is infinitely generated.

Let M $M$ be an orientable connected n $n$ -dimensional manifold with n ∈ { 6 , 7 } $n\in \lbrace 6,7\rbrace$ and let Y ⊂ M $Y\subset M$ be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of M $M$ and Y $Y$ are either both spin or both nonspin. Using Gromov's μ $\mu$

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary s , d $s,d$ the torsion of the homotopy group π s ( S d ) $\pi _s(S^d)$ into a

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander

We provide a sufficient condition for splitness of a link in terms of its reduced sl ( N ) $\mathfrak {sl}(N)$ link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If N $N$ is prime and the coefficient field is of characteristic N $N$ , then the sufficient condition for splitness is also necessary

Given a lattice Veech group in the mapping class group of a closed surface S $S$ , this paper investigates the geometry of Γ $\Gamma$ , the associated π 1 S $\pi _1S$ -extension group. We prove that Γ $\Gamma$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of

We show that the sheaf of A 1 ${\mathbb {A}}^1$ -connected components of a reductive algebraic group over a perfect field is strongly A 1 ${\mathbb {A}}^1$ -invariant. As a consequence, torsors under such groups give rise to A 1 ${\mathbb {A}}^1$ -fiber sequences. We also show that sections of A 1 ${\mathbb {A}}^1$ -connected components of anisotropic, semisimple, simply connected algebraic groups

We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3-manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular

We give an obstruction to positive scalar curvature metrics on 4-manifolds with the homology S 1 × S 3 $S^{1} \times S^{3}$ described in terms of homology cobordism invariants from Seiberg–Witten theory. The main tool of the proof is a relative Bauer–Furuta-type invariant on a periodic-end 4-manifold.

We show that for a closed hyperbolic 3-manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to

We prove that semisimple four-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four-manifolds and homotopy equivalent simply connected closed oriented smooth four-manifolds. We show that all currently known four-dimensional field theories are semisimple, including unitary field theories, and

We prove that the comparison map from G $G$ -fixed points to G $G$ -homotopy fixed points, for the G $G$ -fold smash power of a bounded below spectrum  B $B$ , becomes an equivalence after p $p$ -completion if G $G$ is a finite p $p$ -group and H ∗ ( B ; F p ) $H_*(B; \mathbb {F}_p)$ is of finite type. We also prove that the map becomes an equivalence after I ( G ) $I(G)$ -completion if G $G$ is any

We introduce a type of surgery decomposition of Weinstein manifolds that we call simplicial decompositions. The main result of this paper is that the Chekanov–Eliashberg dg-algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and we show that

We characterize the representations of the fundamental group of a closed surface to ◂⋅▸PSL2⁢(C)$\mathrm{PSL}_2(\mathbb {C})$ that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces

For a Legendrian link ◂⊂▸Λ⊂J1⁢M$\Lambda \subset J^1M$ with M=R$M = \mathbb {R}$ or S1$S^1$, immersed exact Lagrangian fillings L⊂Symp◂≅▸(J1⁢M)≅T∗⁢◂()▸(R>0×M)$L \subset \mbox{Symp}(J^1M) \cong T^*(\mathbb {R}_{>0} \times M)$ of Λ$\Lambda$ can be lifted to conical Legendrian fillings ◂⊂▸Σ⊂J1⁢◂()▸(R>0×M)$\Sigma \subset J^1(\mathbb {R}_{>0} \times M)$ of Λ$\Lambda$. When Σ$\Sigma$ is embedded, using the

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible ◂⋅▸S⁢U⁡(2)$SU(2)$-representations.

The purpose of this paper is to introduce Nk⁡(ℓ)$N_k(\ell )$-maps (◂,▸1⩽k,ℓ⩽∞$1\leqslant k,\ell \leqslant \infty$), which describe higher homotopy normalities, and to study their basic properties and examples. An Nk⁡(ℓ)$N_k(\ell )$-map is defined with higher homotopical conditions. It is shown that a homomorphism is an Nk⁡(ℓ)$N_k(\ell )$-map if and only if there exists fiberwise maps between fiberwise

We provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive

We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups ◂f()▸π∗⁡◂()▸(B⁢Diff∂⁢(Dd))⊗Q$\pi _*(B\mathrm{Diff}_{\partial }(D^d))\otimes {\mathbb {Q}}$ are lifted to homotopy groups of the moduli space of h$h$-cobordisms ◂f()▸π∗⁡◂()

We define the Milnor number of a one-dimensional holomorphic foliation F$\mathcal {F}$ as the intersection number of two holomorphic sections with respect to a compact connected component C$C$ of its singular set. Under certain conditions, we prove that the Milnor number of F$\mathcal {F}$ on a three-dimensional manifold with respect to C$C$ is invariant by C1$C^1$ topological equivalences.

We study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension ◂⩾▸2⁢n+1⩾7$2n +1 \geqslant 7$ with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension d⩾2$d\geqslant

Given an irreducible, end-periodic homeomorphism f:S→S$f: S \rightarrow S$ of a surface with finitely many ends, all accumulated by genus, the mapping torus, Mf$M_f$, is the interior of a compact, irreducible, atoroidal 3-manifold ◂◽.▸M¯f$\overline{M}_f$ with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of ◂◽.▸M¯f$\overline{M}_f$ in terms of the translation

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus g$g$ with respect to the Weil–Petersson measure on the moduli space Mg$\mathcal {M}_g$. We show that as g$g$ goes to infinity, a generic surface X∈Mg$X\in \mathcal {M}_g$ satisfies asymptotically:

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce

We define on any affine invariant orbifold M$\mathcal {M}$ a foliation FM$\mathcal {F}^{\mathcal {M}}$ that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification

The earring tangle consists of four strands 4pt◂⊂▸×I⊂S2×I$4\text{pt} \times I \subset S^2 \times I$ and one meridian around one of the strands. Equipping this tangle with a nontrivial ◂⋅▸S⁢O⁡(3)$SO(3)$ bundle, we show that its traceless ◂⋅▸S⁢U⁡(2)$SU(2)$ flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli

We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.

We prove that if a fibered knot K$K$ with genus greater than 1 in a three-manifold M$M$ has a sufficiently complicated monodromy, then K$K$ induces a minimal genus Heegaard splitting P$P$ that is unique up to isotopy, and small genus Heegaard splittings of M$M$ are stabilizations of P$P$. We provide a complexity bound in terms of the Heegaard genus of M$M$. We also provide global complexity bounds

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy

We show that the stable module ∞$\infty$-category of a finite group G$G$ decomposes in three different ways as a limit of the stable module ∞$\infty$-categories of certain subgroups of G$G$. Analogously to Dwyer's terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex of the graph. This extends previously established results for the pants graph and the separating curve graph to a broad family of graphs associated to surfaces

We use Galois group actions on étale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related objects, to the dg-operad of singular chains of the operad of little disks and to the dg-algebra of singular cochains of the configuration space of points in the complex space. The formality that we obtain is only up to a certain

We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes. We apply the construction to give constraints on the algebraic

We show that the Iwahori–Hecke algebras Hn$\mathcal {H}_n$ of type An−1$A_{n-1}$ satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaoka's homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley–Lieb algebras

For any smooth connected linear algebraic group G$G$ over an algebraically closed field k$k$, we describe the Picard group of the universal moduli stack of principal G$G$-bundles over pointed smooth k$k$-projective curves.

We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential, or infinite. We prove that a RACG is strongly thick of order k$k$ if and only if its divergence function is a polynomial of degree k+1$k+1$. Moreover, we show that the exact divergence function of a RACG can

We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.

We study the elliptic spectral sequence computing tmf∗(RP2)$tmf_*(\mathbb {R}P^2)$ and tmf∗(RP2∧CP2)$tmf_* (\mathbb {R} P^2 \wedge \mathbb {C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, η$\eta$, and ν$\nu$. For tmf∗(RP2∧CP2)$tmf_* (\mathbb {R} P^2 \wedge \mathbb {C} P^2)$, we also compute the effect of the v1$v_1$-self maps of RP2∧CP2$\mathbb {R} P^2 \wedge

We present the first examples of elements in the fundamental group of the space of Legendrian links in (S3,ξst)$(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These new families include the

We study the orientability of vector bundles with respect to a family of cohomology theories called EO$\mathrm{EO}$-theories. The EO$\mathrm{EO}$-theories are higher height analogues of real K$\mathrm{K}$-theory KO$\mathrm{KO}$. For each EO$\mathrm{EO}$-theory, we prove that the direct sum of i$i$ copies of any vector bundle is EO$\mathrm{EO}$-orientable for some specific integer i$i$. Using a splitting

Birman–Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of the Birman–Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. As applications, we give a solution of the braid index problem, the problem to determine the braid index of a given link, and provide estimates of the crossing

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