Showing papers in "Journal of Topology in 2016"

On the algebraic K-theory of higher categories

Abstract: We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.

The median class and superrigidity of actions on CAT(0) cube complexes

Abstract: We define a bounded cohomology class, called the median class, in the second bounded cohomology, with appropriate coefficients, of the automorphism group of a finite-dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the median class of an action by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to what extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show, for example, that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite-dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof, we construct a Γequivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X.

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

Abstract: In [10], we construct an exotic monotone Lagrangian torus in CP 2 (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it T(1,4,25) because, when following a degeneration of CP 2 to the orbifold CP(1,4,25), it degenerates to the central fiber of the moment map for the standard torus action on CP(1,4,25). Related to each degeneration from CP 2 to CP(a 2 ,b 2 ,c 2 ), for (a,b,c) a Markov triple - see (1.1) - there is a monotone Lagrangian torus, which we call T(a 2 ,b 2 ,c 2 ). We conjectured that no two of them are Hamiltonian isotopic to each other. Here we employ techniques from symplectic field theory to prove that the above conjecture is true.

A flexible construction of equivariant Floer homology and applications

Abstract: Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsvath-Szabo, Lipshitz-Treumann, Szabo, Sarkar-Seed-Szabo, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith-type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar-Seed-Szabo's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

Positive scalar curvature and product formulas for secondary index invariants

Abstract: We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher rho-invariant of a metric with uniformly positive scalar curvature as well as for the higher relative index of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense. A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.

Veering triangulations and Cannon–Thurston maps

Abstract: Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.

Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings

Abstract: We define analogues of the graphs of free splittings and of (maximally-)cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of $(G,\{[G_1],\dots,[G_k]\})$-(maximally-)cyclic splittings with the space of equivalence classes of $\mathcal{Z}^{(max)}$-averse trees in the boundary of the corresponding outer space. A tree is \emph{$\mathcal{Z}^{(max)}$-averse} if it is not compatible with any tree $T'$, that is itself compatible with a $(G,\mathcal{F})$-(maximally-)cyclic splitting. Two $\mathcal{Z}^{(max)}$-averse trees are \emph{equivalent} if they are both compatible with a common tree in the boundary of the corresponding outer space.

The L2-Alexander torsion of 3-manifolds

Abstract: We introduce L-2-Alexander torsions for 3-manifolds, which can be viewed as a generalization of the L-2-Alexander invariant of Li-Zhang. We state the L-2-Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that, given any irreducible 3-manifold there exists a coefficient system such that the corresponding L-2-torsion detects the Thurston norm.

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Abstract: Let (E, ∂E , θ) be a stable Higgs bundle of degree 0 on a compact connected Riemann surface. Once we fix a flat metric hdet(E) on the determinant of E, we have the harmonic metrics ht (t > 0) for the stable Higgs bundles (E, ∂E , tθ) such that det(ht) = hdet(E). We study the behaviour of ht when t goes to ∞. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence (E, ∂E , θ, ht) in the case rankE = 2. We introduce a rule to determine the parabolic weights of a “limiting configuration”, and we show the convergence of the sequence to the limiting configuration in an appropriate sense.

The Witt group of real algebraic varieties

Abstract: The purpose of this paper is to compare the algebraic Witt group W (V ) of quadratic forms for an algebraic variety V over R with a new topological invariant, WR(VC), based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of V . This invariant lies between W (V ) and the group KO(VR) of R-linear topological vector bundles on the space VR of real points of V . We show that the comparison maps W (V ) → WR(VC) and WR(VC) → KO(VR) are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for their exponent of the kernel and cokernel, depending upon the dimension of V. These results improve theorems of Knebusch, Mahe and Brumfiel. Along the way, we prove comparison theorem between algebraic and topological hermitianK-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Abstract: Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.

On formality of Sasakian manifolds

Abstract: We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this, we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every n > 3, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension 2n + 1 that are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

Large-scale rank and rigidity of the Teichmüller metric

Abstract: We study the coarse geometry of the Teichmuller space of a compact orientable surface in the Teichmuller metric. We describe when this admits a quasi-isometric embedding of a euclidean space, or a euclidean half-space. We prove quasi-isometric rigidity for Teichmuller space of a surface of complexity at least 2: a result proven independently by Eskin, Masur and Rafi. We deduce that, apart from some well-known coincidences, the Teichmuller spaces are quasi-isometrically distinct. (See also Lemma 2.5 for further discussion.) We also show that Teichmuller space satisfies a quadratic isoperimetric inequality. A key ingredient for proving these results is the fact that Teichmuller space admits a ternary operation, natural up to bounded distance, which endows the space with the structure of a coarse median space whose rank is equal to the complexity of the surface. From this, one can also deduce that any asymptotic cone is bilipschitz equivalent to a CAT(0) space, and so in particular, is contractible

Multiplicity of analytic hypersurface singularities under bi‐Lipschitz homeomorphisms

Abstract: We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.

Knotted surfaces in 4-manifolds and stabilizations

Abstract: In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations - analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.

On the immersion classes of nearby Lagrangians

Abstract: We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immer ...

A note on the concordance of fibered knots

Abstract: Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.

Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree

Abstract: Given a link in the 3-sphere, Ozsvath and Szabo showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of bimodules associated to Dehn twists, and a general pairing theorem for polygons. The previous part (Lipshitz, Ozsvath and Thurston ‘Bordered Floer homology and the spectral sequence of a branched double cover I’, J. Topol. 7 (2014) 1155–1199) focuses on computing the bimodules; this part focuses on the pairing theorem for polygons, in order to prove that the spectral sequence constructed in the previous part agrees with the one constructed by Ozsvath and Szabo.

Shake slice and shake concordant knots

Abstract: A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice knot is one for which a generator of the second homology of this 4-manifold can be represented by a smoothly embedded 2-sphere. It is not known whether there exist 0-shake slice knots that are not slice. We define a relative notion of shake sliceness of knots, which we call shake concordance, which is easily seen to be a generalization of classical concordance, and we give the first examples of knots that are 0-shake concordant but not concordant; these may be chosen to be topologically slice. Additionally, for each r we completely characterize r-shake slice and r-shake concordant knots in terms of concordance and satellite operators. Our characterization allows us to construct new families of possible r-shake slice knots that are not slice.

Non-loose Legendrian spheres with trivial contact homology DGA

Abstract: Loose Legendrian n-submanifolds, n >= 2, were introduced by Murphy ('Loose Legendrian embeddings in high dimensional contact manifolds', Preprint, 2012, arXiv:1201.2245) and proved to be flexibl ...

Algebraic characterization of simple closed curves via Turaev's cobracket

Abstract: The vector space $\V$ generated by the conjugacy classes in the fundamental group of an orientable surface has a natural Lie cobracket $\map{\delta}{\V}{\V\times \V}$. For negatively curved surfaces, $\delta$ can be computed from a geodesic representative as a sum over transversal self-intersection points. In particular $\delta$ is zero for any power of an embedded simple closed curve. Denote by Turaev(k) the statement that $\delta(x^k) = 0$ if and only if the nonpower conjugacy class $x$ is represented by an embedded curve. Computer implementation of the cobracket delta unearthed counterexamples to Turaev(1) on every surface with negative Euler characteristic except the pair of pants. Computer search have verified Turaev(2) for hundreds of millions of the shortest classes. In this paper we prove Turaev(k) for $k=3,4,5,\dots$ for surfaces with boundary. Turaev himself introduced the cobracket in the 80's and wondered about the relation with embedded curves, in particular asking if Turaev (1) might be true. We give an application of our result to the curve complex. We show that a permutation of the set of free homotopy classes that commutes with the cobracket and the power operation is induced by an element of the mapping class group.

Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4n+1

Abstract: Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold $M$, on the linking form associated to the homology groups of $M$. In particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theorem, we prove several disjunction results for the embeddings and immersions of $\mathbb{Z}/k$-manifolds that could be of independent interest.

Torsion homology and regulators of isospectral manifolds

Abstract: Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg_i(M_1)/Reg_i(M_2) representation theoretically. Further, we prove that the p-primary torsion in the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing #G. For p <= 71, we give examples of pairs of isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.

Solving Thurston's equation in a commutative ring

Abstract: We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.

Characterization of finite groups generated by reflections and rotations

Abstract: We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.

Rigidity and characteristic classes of smooth bundles with nonpositively curved fibers

Abstract: We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.

Exact triangles for SO(3) instanton homology of webs

Abstract: The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or “web”). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. From the octahedral diagram, one can derive equivalent reformulations of the authors' conjecture that, for planar webs, the rank of the instanton homology is equal to the number of Tait colorings.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

Abstract: We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

Bounds on genus and configurations of embedded surfaces in 4-manifolds

Abstract: For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction inequalities for surfaces embedded in $m\mathbb{CP}^2\# n(-\mathbb{CP}^2)$ ($m, n \geq 2$). The proofs of these results are given by studying a family of Seiberg-Witten equations.