UCLA Topology Group

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The homotopy category of $N_\infty$ operads is equivalent to a finite lattice, and as the ambient group varies, there are various image constructions between these lattices. In this paper, we explain how to lift this algebraic structure back to the operad level. We show that lattice joins and meets correspond to derived operadic coproducts and products, and we show that the image constructions correspond to derived operadic induction, restriction, and coinduction, at least when taken along an injective homomorphism. We also prove that a derived variant of the Boardman-Vogt tensor product lifts the join. Our result does not resolve Blumberg and Hill's conjecture on the usual tensor product, but it does imply that every $N_\infty$ ring spectrum can be replaced with an equivalent spectrum, which is equipped with a self-interchanging operad action.

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We study how a smashing Bousfield localization behaves under various equivariant functors. We show that the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ do not determine smashing localizations except when $n=0$, and we establish a version of the chromatic convergence theorem for the $L_{E_{\mathbb{R}}(n)}$ chromatic tower. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear.

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In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories are built from the categories of $\mathcal{O}$-algebras in $G$-spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties of the equivariant stable category, notably the tom Dieck splitting. Our work is motivated in part by the examples arising from the equivariant units and Picard space functors.

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We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces.

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We generalize the mixed tori which appear in the second author's JSJ-type decomposition theorem for symplectic fillings of contact manifolds.

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Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor $J^\sharp$, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement $J^\flat$ for $J^\sharp$. Of particular interest is the relationship between the dimension of $J^\flat(K)$ for a web $K$ and the number of Tait colorings $\Tait(K)$ of $K$; these two numbers are known to be identical for a special class of "reducible"' webs, but whether this is the case for nonreducible webs is not known. We describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of $J^\flat(K)$ for a given web $K$, in some cases determining these quantities uniquely. We present results for a number of nonreducible example webs. For the dodecahedral web $W_1$ the number of Tait colorings is $\Tait(W_1) = 60$, but our results suggest that $\dim J^\flat(W_1) = 58$.

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We prove the C^0-genericity of convex hypersurfaces. There are lots of interesting consequences of the technology introduced, some of which are proven in the paper and others which will be in future papers.

Algebraic and Geometric Topol., to appear
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We use orders of certain torsion elements in Khovanov homology to produce lower bounds on complexities of ribbon concordances between knots.

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We generalize to higher dimensions some properties of surface automorphisms of pseudo-Anosov type.

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We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill's horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and we develop basic tools for computing with them.

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We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height $2$ Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order $32$ generated by $E\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\wedge S^{7+\sigma}$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum $E$.

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Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a finite symplectic group action and satisfy certain simplifying assumptions.

J. Differential Geom., to appear
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We survey the interactions between foliations and contact structures in dimension three, with an emphasis on sutured manifolds and invariants of sutured contact manifolds. This paper contains two original results: the fact that a closed orientable irreducible 3-manifold M with nonzero second homol-ogy carries a hypertight contact structure and the fact that an orientable, taut, balanced sutured 3-manifold is not a product if and only if it carries a contact structure with nontrivial cylindrical contact homology.

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We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C4))}\langle 2\rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E^{hC_{12}}_4$ is 384-periodic.

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We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted arc, as well as holonomy perturbations of the Chern-Simons functional used to define the homology theory. The other solid torus contains the remainder of the knot. The Heegaard splitting yields a pair of Lagrangians in the traceless $SU(2)$-character variety of the twice-punctured torus, and the intersection points of these Lagrangians comprise the generating set that we seek. We illustrate the scheme by constructing generating sets for several example knots. Our scheme is a direct generalization of a scheme introduced by Hedden, Herald, and Kirk for describing generating sets for knots in $S^3$ in terms of Lagrangian intersections in the traceless $SU(2)$-character variety for the 2-sphere with four punctures.

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We introduce a computationally tractable way to describe the $\mathbb Z$-homotopy fixed points of a $C_{n}$-spectrum $E$, producing a genuine $C_{n}$ spectrum $E^{hn\mathbb Z}$ whose fixed and homotopy fixed points agree and are the $\mathbb Z$-homotopy fixed points of $E$. These form a piece of a contravariant functor from the divisor poset of $n$ to genuine $C_{n}$-spectra, and when $E$ is an $N_{\infty}$-ring spectrum, this functor lifts to a functor of $N_{\infty}$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $E^{hn\mathbb Z}$, giving the homotopy groups of the $\mathbb Z$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $\mathbb Z$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.

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We prove a JSJ-type decomposition theorem for strong and exact symplectic fillings of a contact 3-manifold when it is cut along a certain convex torus called a mixed torus. As an application we show the uniqueness of exact fillings when the contact manifold is obtained by Legendrian surgery on a knot in S^3 when the knot is stabilized both positively and negatively.

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We propose definitions of complex manifolds ${\cal H}(T^2,n)$ for $n \in \Nats$ that could potentially be used to construct the symplectic Khovanov homology of $n$-stranded links in lens spaces. The space ${\cal H}(T^2,n)$ is defined in terms of Hecke modifications of rank 2 vector bundles over an elliptic curve. We undertake a detailed study of such Hecke modifications: we show which vector bundles can Hecke modify to which, and in what directions. We also describe explicit morphisms of vector bundles that represent all possible Hecke modifications of all possible rank 2 vector bundles on elliptic curves. Using these results, we explicitly characterize the space ${\cal H}(T^2,n)$ for $n = 0, 1, 2$, and we describe an embedding of ${\cal H}(T^2,n)$ into the moduli space $M^{ss}(T^2,n+1)$ of rank 2 semistable parabolic bundles. For comparison, we present analogous results for the case of rational curves, for which the corresponding space of Hecke modifications ${\cal H}(S^2,n)$ is isomorphic to a space ${\cal Y}(S^2,n)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$-stranded links in $S^3$.

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This is in some sense a sequel of "Convex hypersurface theory in contact topology", although written earlier. We generalize the bypass attachment to arbitrary dimensions. We also introduce a new type of overtwisted object called the overtwisted orange which is middle-dimensional and contractible.

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We produce a spatial refinement of odd Khovanov homology, similar in spirit to the spatial refinement of even Khovanov homology, but using functors from a signed version of the Burnside category.

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The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron $X$ is extended to n-valued maps by replacing liftings to the universal covering space by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl.

Proc. of the ICM
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We review the construction and context of a stable homotopy refinement of Khovanov homology.

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We describe the structure present in algebras over the little disks operads for various representations of a finite group $G$, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_2$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.

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Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which $S^n$ has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which $S^n$ has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.

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We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.

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For G a Lie group acting on a symplectic manifold preserving a pair of Lagrangians L0, L1, under certain hypotheses not including equivariant transversality we construct a G-equivariant Floer cohomology of L0 and L1.

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We prove in general that contact homology is defined.