Topological research at UCLA began with the arrival of Robert Sorgenfrey in 1942. High points in the research accomplishments of topologists at UCLA include the solution by
Robion Kirby, who was at UCLA from 1965 to 1971, (with Laurence Siebenmann) of four of the seven problems listed by John Milnor in 1963 as the most important in topology at that
time. Kirby first presented his famous torus trick, the key to the solutions, in a UCLA seminar in the summer of 1968. Another of the Milnor problems, the Double Suspension
Conjecture was solved by Robert Edwards, who came to UCLA in 1970 and remained here until his retirement in 2006. The accomplishments of Allan Hatcher, who was at UCLA from
1976 until 1984, include the proof of the Smale Conjecture (published in 1983).

There is
a Google
group mailing list that announces topology seminars occuring in
UCLA; to be included in the mailing list, apply through the Google
group webpage, or
contact Mike
Hill, Ko Honda,
or Sucharit
Sarkar. The following seminars occur in or around UCLA.

- Joint UCLA / USC / Caltech Topology Seminar
- USC Geometry, Topology, and Categorification Seminar
- Caltech Geometry and Topology Seminar

Here is a list of some recent topology workshops (co-)organized by UCLA topologists.

- Equivariant Techniques in Stable Homotopy (May 2020)
- Equivariant Stable Homotopy and p-adic Hodge Theory (March 2020)
- Geometry and Topology Workshop (January 2020)
- Floer Homotopy Theory and Low-dimensional Topology Workshop (August 2019)
- Workshop on Homotopy Theory (August 2019)
- QFT and Manifold Invariants (July 2019)
- Hidden Algebraic Structures in Topology (March 2019).
- Symplectic Geometry and Homotopy Theory Workshop (December 2018)
- Chromatic Homotopy Theory, Journey to the Frontier (May 2018)
- Low-dimensional Topology Workshop (January 2018)
- Winter School in Algebraic Topology (October 2017)
- Floer Homology and Homotopy Theory Workshop (July 2017)
- Gauge Theory and Categorification (March 2017)

Here is a list of recent papers by UCLA topologists; click on the paper to see details.

## (Carrick) Cofreeness in Real bordism theory and the Segal conjecture

ArXiv

We prove that the genuine $C_{2^n}$-spectrum $N_{C_2}^{C_{2^n}}MU_{\mathbb{R}}$ is cofree, for all $n$. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal conjecture for $C_2$, independent of Lin's theorem.

We prove that the genuine $C_{2^n}$-spectrum $N_{C_2}^{C_{2^n}}MU_{\mathbb{R}}$ is cofree, for all $n$. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal conjecture for $C_2$, independent of Lin's theorem.

## (Akhmechet-Krushkal-Willis) Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum

ArXiv

We lift the actions of the standard generators of $\mathfrak{sl}_2$ on annular Khovanov homology to maps on annular Khovanov spectra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

We lift the actions of the standard generators of $\mathfrak{sl}_2$ on annular Khovanov homology to maps on annular Khovanov spectra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

## (Brown) On the Nielsen root theory of n-valued maps

An n-value map between closed triangulated manifolds of the same
dimension is homotopic to a map with finitely many root. We conjecture that it is homotopic to a map with the Nielsen root number of roots except for surfaces and we prove the
conjecture for n-valued self-maps of Lie groups. We calculate the Nielsen root numbers of linear n-valued maps of tori.

## (Christian-Li) Some applications of Menke's JSJ decomposition for symplectic fillings

ArXiv

We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces.

We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces.

## (Akhmechet-Krushkal-Willis) Stable homotopy refinement of quantum annular homology

Compos. Math.

ArXiv

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar.

ArXiv

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar.

## (Manolescu-Marengon-Sarkar-Willis) A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds

ArXiv

We generalize Rasmussen's s-invariant for certain links inside connected sums of S^1\times S^2's, and prove adjunction inequality for s-invariant in CP^2. As a corollary, we prove that the Freedman-Gompf-Morrison-Walker strategy cannot be used to prove that Gluck twists are not diffeomorphic to the 4-ball.

We generalize Rasmussen's s-invariant for certain links inside connected sums of S^1\times S^2's, and prove adjunction inequality for s-invariant in CP^2. As a corollary, we prove that the Freedman-Gompf-Morrison-Walker strategy cannot be used to prove that Gluck twists are not diffeomorphic to the 4-ball.

## (Lipshitz-Sarkar) Khovanov homology also detects split links

ArXiv

We prove that the module structure on Khovanov homology detects split links.

We prove that the module structure on Khovanov homology detects split links.

## (Lawson-Lipshitz-Sarkar) Chen-Khovanov spectra for tangles

ArXiv

We define stable homotopy refinements of Chen-Khovanov algebras and tangle invariants.

We define stable homotopy refinements of Chen-Khovanov algebras and tangle invariants.

## (Rubin) Categorifying the algebra of indexing systems

ArXiv

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.

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.

## (Carrick) Smashing localizations in equivariant stable homotopy

ArXiv

We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of $E_\infty$-rings.

We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of $E_\infty$-rings.

## (Blumberg-Hill) Equivariant stable categories for incomplete systems of transfers

ArXiv

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.

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.

## (Christian) On symplectic fillings of virtually overtwisted torus bundles

Algebr. Geom. Topol.

ArXiv

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.

ArXiv

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.

## (Christian-Menke) Splitting symplectic fillings

ArXiv

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

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

## (Boozer) Computer bounds for Kronheimer-Mrowka foam evaluation

ArXiv

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 $\dim J^\flat(W_1) = 58$.

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 $\dim J^\flat(W_1) = 58$.

## (Honda-Huang) Convex hypersurface theory in contact topology

ArXiv

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.

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.

## (Willis) Khovanov-Rozansky homology for infinite multi-colored braids

Canad. J. Math.

ArXiv

We define a limiting sl_N Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids.

ArXiv

We define a limiting sl_N Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids.

## (Sarkar) Ribbon distance and Khovanov homology

Algebr. Geom. Topol.

ArXiv

We use orders of certain torsion elements in Khovanov homology to produce lower bounds on complexities of ribbon concordances between knots.

ArXiv

We use orders of certain torsion elements in Khovanov homology to produce lower bounds on complexities of ribbon concordances between knots.

## (Lee) Towards a higher-dimensional construction of stable/unstable Lagrangian laminations

ArXiv

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

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

## (Rubin) Characterizations of equivariant Steiner and linear isometries operads

ArXiv

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.

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.

## (Manion-Marengon-Willis) Strands algebras and Ozsváth-Szabó's Kauffman-states functor

ArXiv

We construct "strands algebras" following a construction from bordered Floer homology, and we prove that they are quasi-isomorphic to Ozsváth-Szabó's "bordered algebras".

We construct "strands algebras" following a construction from bordered Floer homology, and we prove that they are quasi-isomorphic to Ozsváth-Szabó's "bordered algebras".

## (Manion-Marengon-Willis) Generators, relations, and homology for Ozsváth-Szabó's Kauffman-states algebras

ArXiv

We give a presentation of Ozsváth-Szabó's bordered dg algebras as quotients of path algebras. We also determine their homology and their formality.

We give a presentation of Ozsváth-Szabó's bordered dg algebras as quotients of path algebras. We also determine their homology and their formality.

## (Hill) Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra

ArXiv

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 $E$.

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 $E$.

## (Bao-Honda) Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures

ArXiv

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.

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.

## (Willis) Khovanov homology for links in #^r(S^2 \times S^1)

Michigan Math. J.

ArXiv

We revisit Rozansky's construction of Khovanov homology for links in S2xS1 and extend it to connect sums. We also explore knotifications and decategorifications.

ArXiv

We revisit Rozansky's construction of Khovanov homology for links in S2xS1 and extend it to connect sums. We also explore knotifications and decategorifications.

## (Colin-Honda) Foliations, contact structures and their interactions in dimension three

J. Differential Geom.

ArXiv

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.

ArXiv

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.

## (Hill-Shi-Wang-Xu) The slice spectral sequence of a $C_4$-equivariant height-4 Lubin-Tate theory

ArXiv

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.

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.

## (Boozer) Holonomy perturbations of the Chern-Simons functional for lens spaces

ArXiv

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.

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.

## (Manolescu-Marengon) The Knight Move Conjecture is false

Proc. Amer. Math. Soc.

ArXiv

The Knight Move Conjecture in Khovanov homology says that the Khovanov homology of a knot decomposes as the direct sum of two kinds of elementary pieces. We exhibit a counterexample to this conjecture.

ArXiv

The Knight Move Conjecture in Khovanov homology says that the Khovanov homology of a knot decomposes as the direct sum of two kinds of elementary pieces. We exhibit a counterexample to this conjecture.

## (Hill-Zeng) The $\mathbb Z$-homotopy fixed points of $C_{n}$ spectra with applications to norms of $MU_{\mathbb R}$

ArXiv

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.

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.

## (Menke) A JSJ-type decomposition theorem for symplectic fillings

ArXiv

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.

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.

## (Boozer) Moduli spaces of Hecke modifications for rational and elliptic curves

ArXiv

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 $S^3$.

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 $S^3$.

## (Honda-Huang) Bypass attachments in higher-dimensional contact topology

ArXiv

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.

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.

## (Sarkar-Scaduto-Stoffregen) An odd Khovanov homotopy type

ArXiv

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.

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.

## (Brown-Deconinck-Dekimpe-Staecker) Lifting classes for the fixed point theory of n-valued maps

Topology Appl.

ArXiv

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.

ArXiv

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.

## (Abel-Willis) Colored Khovanov-Rozansky homology for infinite braids

Algebr. Geom. Topol.

ArXiv

We show that the limiting unicolored sl_N chain complex of any infinite positive briad categorifies a highest-weight projector. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp.

ArXiv

We show that the limiting unicolored sl_N chain complex of any infinite positive briad categorifies a highest-weight projector. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp.

## (Lipshitz-Sarkar) Spatial refinements and Khovanov homology

Proc. of the ICM

ArXiv

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

ArXiv

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

## (Hill) On the algebras over equivariant little disks

ArXiv

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.

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.

## (Behrens-Hill-Hopkins-Mahowald) Detecting exotic spheres in low dimensions using coker J

ArXiv

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.

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.

## (Lawson-Lipshitz-Sarkar) Khovanov spectra for tangles

ArXiv

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

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

## (Hendricks-Lipshitz-Sarkar) A simplicial construction of G-equivariant Floer homology

ArXiv

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.

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.

## (Bao-Honda) Semi-global Kuranishi charts and the definition of contact homology

ArXiv

We prove in general that contact homology is defined.

We prove in general that contact homology is defined.