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.
 Spectral Methods in Algebra, Geometry, and Topology (September 2022  December 2022)
 Equivariant Techniques in Stable Homotopy (May 2021 / September 2022)
 Equivariant Stable Homotopy and padic Hodge Theory (March 2020)
 Geometry and Topology Workshop (January 2020)
 Floer Homotopy Theory and Lowdimensional 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)
 Lowdimensional 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.
(AngeliniKnollGerhardtHill) Real topological Hochschild homology via the norm and Real Witt vectors
ArXiv
We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with antiinvolution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$typical Witt vectors of rings with antiinvolution. We end with an explicit computation of the degree zero $D_{2m}$Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.
We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with antiinvolution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$typical Witt vectors of rings with antiinvolution. We end with an explicit computation of the degree zero $D_{2m}$Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.
(HillLawson) $E_k$pushouts and $E_{k+1}$tensors
ArXiv
We prove a general result that relates certain pushouts of $E_k$algebras to relative tensors over $E_{k+1}$algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology theories for ring spectra. The main results apply when the category in question has centralizers. Among our applications, we show that certain quotients of the dual Steenrod algebra are realized as associative algebras over $HF_p \wedge HF_p$ by attaching single $E_1$algebra relation, generalizing previous work at the prime $2$. We also construct a filtered $E_2$algebra structure on the sphere spectrum, and the resulting spectral sequence for the stable homotopy groups of spheres has $E_1$term isomorphic to a regrading of the $E_1$term of the May spectral sequence.
We prove a general result that relates certain pushouts of $E_k$algebras to relative tensors over $E_{k+1}$algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology theories for ring spectra. The main results apply when the category in question has centralizers. Among our applications, we show that certain quotients of the dual Steenrod algebra are realized as associative algebras over $HF_p \wedge HF_p$ by attaching single $E_1$algebra relation, generalizing previous work at the prime $2$. We also construct a filtered $E_2$algebra structure on the sphere spectrum, and the resulting spectral sequence for the stable homotopy groups of spheres has $E_1$term isomorphic to a regrading of the $E_1$term of the May spectral sequence.
(LipshitzSarkar) A mixed invariant of nonorientable surfaces in equivariant Khovanov homology
ArXiv
We construct a mixed invariant of nonorientable surfaces from the Lee and BarNatan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including some exotic examples.
We construct a mixed invariant of nonorientable surfaces from the Lee and BarNatan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including some exotic examples.
(ManolescuSarkar) A knot Floer stable homotopy type
ArXiv
Given a grid diagram for a knot or link K in $S^3$, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudoholomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions.
Given a grid diagram for a knot or link K in $S^3$, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudoholomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions.
(HillMehrleQuigley) Free incomplete Tambara functors are almost never flat
ArXiv
Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.
Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.
(LawsonLipshitzSarkar) Homotopy functoriality for Khovanov spectra
ArXiv
We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.
We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.
(BlumbergHill) Biincomplete Tambara functors
ArXiv
For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If $R$ is an $N_\infty$ ring spectrum in the category of genuine $G$spectra, then all possible additive transfers are present and $\pi_0 R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_\infty$ ring spectrum in a category of incomplete $G$spectra, the situation is more subtle. In this paper, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call biincomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, biincomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.
For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If $R$ is an $N_\infty$ ring spectrum in the category of genuine $G$spectra, then all possible additive transfers are present and $\pi_0 R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_\infty$ ring spectrum in a category of incomplete $G$spectra, the situation is more subtle. In this paper, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call biincomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, biincomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.
(BeaudryHillLawsonShiZeng) Quotient rings of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$
ArXiv
We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$algebras whose coefficient rings are finitedimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.
We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$algebras whose coefficient rings are finitedimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.
(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.
(AkhmechetKrushkalWillis) Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum
ArXiv
Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. 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.
Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. 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.
(ChristianLi) 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 3manifolds. Among other results, we complete the classification up to orientationpreserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.
We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3manifolds. Among other results, we complete the classification up to orientationpreserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.
(BeaudryHillShiZeng) Transchromatic extensions in motivic and Real bordism
ArXiv
We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "redshifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $\pi_{(\ast,\ast)}MGL$module structure of the motivic Morava $K$theories, including nontrivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $\pi_{\star}MU_\mathbb R$module structure on the Real Morava $K$theories.
We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "redshifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $\pi_{(\ast,\ast)}MGL$module structure of the motivic Morava $K$theories, including nontrivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $\pi_{\star}MU_\mathbb R$module structure on the Real Morava $K$theories.
(BeaudryHillShiZeng) Models of LubinTate spectra via Real bordism theory
ArXiv
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$equivariant Real oriented models of LubinTate spectra $E_h$ at heights $h=2^{n1}m$ and give explicit formulas of the $C_{2^n}$action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the HillHopkinsRavenel norms of $MU_{\mathbb{R}}$.
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$equivariant Real oriented models of LubinTate spectra $E_h$ at heights $h=2^{n1}m$ and give explicit formulas of the $C_{2^n}$action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the HillHopkinsRavenel norms of $MU_{\mathbb{R}}$.
(AkhmechetKrushkalWillis) Stable homotopy refinement of quantum annular homology
ArXiv
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.
(ManolescuMarengonSarkarWillis) A generalization of Rasmussen's invariant, with applications to surfaces in some fourmanifolds
ArXiv
We extend the definition of KhovanovLee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussentype invariant for nullhomologous links in these manifolds. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussentype invariant to the genus of surfaces with boundary in the following fourmanifolds: $B^2 \times S^2$, $S^1 \times B^3$, $\mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4balls are nonstandard.
We extend the definition of KhovanovLee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussentype invariant for nullhomologous links in these manifolds. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussentype invariant to the genus of surfaces with boundary in the following fourmanifolds: $B^2 \times S^2$, $S^1 \times B^3$, $\mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4balls are nonstandard.
(LipshitzSarkar) Khovanov homology detects split links
ArXiv
Extending ideas of HeddenNi, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is welldefined up to quasiisomorphism.
Extending ideas of HeddenNi, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is welldefined up to quasiisomorphism.
(Hill) Freeness and equivariant stable homotopy
ArXiv
We introduce a notion of freeness for $RO$graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$homology of $E$ splits as a wedge of the $R$homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by HillHopkinsRavenel. For these spectra, the $RO$graded Bredon homology is extremely easy to compute, and if these spaces have additional structure, then this can also be easily determined. In particular, the homology of a space with this property naturally has the structure of a coTambara functor (and compatibly with any additional product structure). We work this out in the example of $BU_{\mathbb R}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various $E_{\infty}$ pushouts, and we apply this to describe the homology of $BBU_{\mathbb R}$.
We introduce a notion of freeness for $RO$graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$homology of $E$ splits as a wedge of the $R$homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by HillHopkinsRavenel. For these spectra, the $RO$graded Bredon homology is extremely easy to compute, and if these spaces have additional structure, then this can also be easily determined. In particular, the homology of a space with this property naturally has the structure of a coTambara functor (and compatibly with any additional product structure). We work this out in the example of $BU_{\mathbb R}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various $E_{\infty}$ pushouts, and we apply this to describe the homology of $BBU_{\mathbb R}$.
(LawsonLipshitzSarkar) ChenKhovanov spectra for tangles
ArXiv
We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of ChenKhovanov and Stroppel's platform algebras and tangle invariants, and discuss the topological Hochschild homology of these refinements.
We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of ChenKhovanov and Stroppel's platform algebras and tangle invariants, and discuss the topological Hochschild homology of these refinements.
(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 BoardmanVogt 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 selfinterchanging 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 BoardmanVogt 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 selfinterchanging 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 JohnsonWilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$equivariant JohnsonWilson 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 quasiGalois 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 JohnsonWilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$equivariant JohnsonWilson 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 quasiGalois extension of $E_\infty$rings.
(BlumbergHill) 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
ArXiv
We use Menke's JSJtype 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. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.
We use Menke's JSJtype 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. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.
(ChristianMenke) Splitting symplectic fillings
ArXiv
We generalize the mixed tori which appear in the second author's JSJtype decomposition theorem for symplectic fillings of contact manifolds. Mixed tori are convex surfaces in contact manifolds which may be used to decompose symplectic fillings. We call our more general surfaces splitting surfaces, and show that the decomposition of symplectic fillings continues to hold. Specifically, given a strong or exact symplectic filling of a contact manifold which admits a splitting surface, we produce a new symplectic manifold which strongly or exactly fills its boundary, and which is related to the original filling by Liouville surgery.
We generalize the mixed tori which appear in the second author's JSJtype decomposition theorem for symplectic fillings of contact manifolds. Mixed tori are convex surfaces in contact manifolds which may be used to decompose symplectic fillings. We call our more general surfaces splitting surfaces, and show that the decomposition of symplectic fillings continues to hold. Specifically, given a strong or exact symplectic filling of a contact manifold which admits a splitting surface, we produce a new symplectic manifold which strongly or exactly fills its boundary, and which is related to the original filling by Liouville surgery.
(Boozer) Computer Bounds for KronheimerMrowka 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 $\mathrm{Tait}(K)$ 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 $\mathrm{Tait}(W_1) = 60$, 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 $\mathrm{Tait}(K)$ 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 $\mathrm{Tait}(W_1) = 60$, but our results suggest that $\dim J^\flat(W_1) = 58$.
(HondaHuang) Convex hypersurface theory in contact topology
ArXiv
We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$approximated by a convex one. We also prove that a $C^0$generic family of mutually disjoint closed hypersurfaces parametrized by $t \in [0,1]$ is convex except at finitely many times $t_1, \dots, t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As applications of CHT, we prove the existence of compatible (relative) open book decompositions for contact manifolds and an existence hprinciple for codimension 2 contact submanifolds.
We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$approximated by a convex one. We also prove that a $C^0$generic family of mutually disjoint closed hypersurfaces parametrized by $t \in [0,1]$ is convex except at finitely many times $t_1, \dots, t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As applications of CHT, we prove the existence of compatible (relative) open book decompositions for contact manifolds and an existence hprinciple for codimension 2 contact submanifolds.
(Willis) KhovanovRozansky homology for infinite multicolored braids
ArXiv
We define a limiting $\mathfrak{sl}_N$ KhovanovRozansky homology for semiinfinite positive multicolored braids, and we show that this limiting homology categorifies a highestweight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semiinfinite and biinfinite multicolored braids.
We define a limiting $\mathfrak{sl}_N$ KhovanovRozansky homology for semiinfinite positive multicolored braids, and we show that this limiting homology categorifies a highestweight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semiinfinite and biinfinite multicolored braids.
(Sarkar) Ribbon distance and Khovanov homology
ArXiv
We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the Xaction on Lee's perturbation of Khovanov homology.
We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the Xaction on Lee's perturbation of Khovanov homology.
(Lee) Towards a higherdimensional construction of stable/unstable Lagrangian laminations
ArXiv
We generalize some properties of surface automorphisms of pseudoAnosov type. First, we generalize the Penner construction of a pseudoAnosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction has an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination, which are higherdimensional generalizations of a train track and a geodesic lamination in the surface case. Moreover, if a pair consisting of a symplectic automorphism $\psi$ and a Lagrangian branched surface $B_{\psi}$ satisfies some assumptions, we prove that there is an invariant Lagrangian lamination $\mathcal{L}$ which is a higherdimensional generalization of a geodesic lamination.
We generalize some properties of surface automorphisms of pseudoAnosov type. First, we generalize the Penner construction of a pseudoAnosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction has an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination, which are higherdimensional generalizations of a train track and a geodesic lamination in the surface case. Moreover, if a pair consisting of a symplectic automorphism $\psi$ and a Lagrangian branched surface $B_{\psi}$ satisfies some assumptions, we prove that there is an invariant Lagrangian lamination $\mathcal{L}$ which is a higherdimensional generalization of a geodesic lamination.
(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 hornfilling 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 hornfilling 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.
(ManionMarengonWillis) Strands algebras and OzsváthSzabó's Kauffmanstates functor
ArXiv
We define new differential graded algebras A(n,k,S) in the framework of LipshitzOzsv\'athThurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'athSzab\'o's algebras B(n,k,S); indeed, we exhibit a quasiisomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'athSzab\'o's gradings on B(n,k,S) arise naturally from the general framework of groupvalued gradings on strands algebras.
We define new differential graded algebras A(n,k,S) in the framework of LipshitzOzsv\'athThurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'athSzab\'o's algebras B(n,k,S); indeed, we exhibit a quasiisomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'athSzab\'o's gradings on B(n,k,S) arise naturally from the general framework of groupvalued gradings on strands algebras.
(ManionMarengonWillis) Generators, relations, and homology for OzsváthSzabó's Kauffmanstates algebras
ArXiv
We give a generatorsandrelations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
We give a generatorsandrelations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
(BeaudryBobkovaHillStojanoska) 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 $E$.
(BaoHonda) Equivariant Lagrangian Floer cohomology via semiglobal Kuranishi structures
ArXiv
Using a simplified version of Kuranishi perturbation theory that we call semiglobal 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 semiglobal 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 $\#^r(S^2\times S^1)$
ArXiv
We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be used to recover WRT invariants at certain roots of unity, and also recovers the evaluation of $L$ in the skein module $\mathcal{S}(M^r)$ of Hoste and Przytycki when $L$ is nullhomologous in $M^r$. The construction also allows for a clear path towards defining a Lee's homology $Kh'(L)$ and associated $s$invariant for such $L$, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in $S^3$ and show directly that this is invariant under handleslides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.
We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be used to recover WRT invariants at certain roots of unity, and also recovers the evaluation of $L$ in the skein module $\mathcal{S}(M^r)$ of Hoste and Przytycki when $L$ is nullhomologous in $M^r$. The construction also allows for a clear path towards defining a Lee's homology $Kh'(L)$ and associated $s$invariant for such $L$, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in $S^3$ and show directly that this is invariant under handleslides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.
(ColinHonda) Foliations, contact structures and their interactions in dimension three
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 3manifold M with nonzero second homology carries a hypertight contact structure and the fact that an orientable, taut, balanced sutured 3manifold is not a product if and only if it carries a contact structure with nontrivial cylindrical contact homology. The proof of the second statement uses the HandelMiller theory of endperiodic diffeomorphisms of endperiodic surfaces.
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 3manifold M with nonzero second homology carries a hypertight contact structure and the fact that an orientable, taut, balanced sutured 3manifold is not a product if and only if it carries a contact structure with nontrivial cylindrical contact homology. The proof of the second statement uses the HandelMiller theory of endperiodic diffeomorphisms of endperiodic surfaces.
(HillShiWangXu) The slice spectral sequence of a $C_4$equivariant height4 LubinTate theory
ArXiv
We completely compute the slice spectral sequence of the $C_4$spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$localization, this spectrum is equivalent to a height4 LubinTate theory $E_4$ with $C_4$action induced from the GoerssHopkinsMiller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384periodic.
We completely compute the slice spectral sequence of the $C_4$spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$localization, this spectrum is equivalent to a height4 LubinTate theory $E_4$ with $C_4$action induced from the GoerssHopkinsMiller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384periodic.
(Boozer) Holonomy perturbations of the ChernSimons 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 Heegaardsplitting 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 ChernSimons 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 twicepunctured 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 2sphere 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 Heegaardsplitting 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 ChernSimons 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 twicepunctured 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 2sphere with four punctures.
(ManolescuMarengon) The Knight Move Conjecture is false
ArXiv
The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov homology to Q^2 converges on the second page, as it does for all alternating knots and knots with unknotting number at most 2. We present a counterexample to the Knight Move Conjecture. For this knot, the Lee spectral sequence admits a nontrivial differential of bidegree (1,8).
The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov homology to Q^2 converges on the second page, as it does for all alternating knots and knots with unknotting number at most 2. We present a counterexample to the Knight Move Conjecture. For this knot, the Lee spectral sequence admits a nontrivial differential of bidegree (1,8).
(HillZeng) 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 JohnsonWilson 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 JohnsonWilson 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 JSJtype decomposition theorem for symplectic fillings
ArXiv
Let $(M,\xi)$ be a contact 3manifold and $T^2 \subset (M,\xi)$ a convex torus of a special type called a mixed torus. We prove a JSJtype decomposition theorem for strong and exact symplectic fillings of $(M,\xi)$ when $(M,\xi)$ is cut along $T^2$. As an application we show the uniqueness of exact fillings when $(M,\xi)$ is obtained by Legendrian surgery on a knot in $(S^3,\xi_{std})$ when the knot is stabilized both positively and negatively.
Let $(M,\xi)$ be a contact 3manifold and $T^2 \subset (M,\xi)$ a convex torus of a special type called a mixed torus. We prove a JSJtype decomposition theorem for strong and exact symplectic fillings of $(M,\xi)$ when $(M,\xi)$ is cut along $T^2$. As an application we show the uniqueness of exact fillings when $(M,\xi)$ is obtained by Legendrian surgery on a knot in $(S^3,\xi_{std})$ 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 $\mathcal{P}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$stranded links in lens spaces. The manifolds $\mathcal{P}_M(X,m,n)$ are defined as moduli spaces of Hecke modifications of rank 2 parabolic bundles over an elliptic curve $X$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank 2 vector bundles over $X$, and we use these results to define a canonical open embedding of $\mathcal{P}_M(X,m,n)$ into $M^s(X,m+n)$, the moduli space of stable rank 2 parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute $\mathcal{P}_M(X,1,n)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold $\mathcal{P}_M(\mathbb{CP}^1,3,n)$ is isomorphic for $n$ even to a space $\mathcal{Y}(S^2,n)$ 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 $\mathcal{P}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$stranded links in lens spaces. The manifolds $\mathcal{P}_M(X,m,n)$ are defined as moduli spaces of Hecke modifications of rank 2 parabolic bundles over an elliptic curve $X$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank 2 vector bundles over $X$, and we use these results to define a canonical open embedding of $\mathcal{P}_M(X,m,n)$ into $M^s(X,m+n)$, the moduli space of stable rank 2 parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute $\mathcal{P}_M(X,1,n)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold $\mathcal{P}_M(\mathbb{CP}^1,3,n)$ is isomorphic for $n$ even to a space $\mathcal{Y}(S^2,n)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$stranded links in $S^3$.
(HondaHuang) Bypass attachments in higherdimensional contact topology
ArXiv
We initiate a systematic study of convex hypersurface theory and generalize the bypass attachment to arbitrary dimensions. We also introduce a new type of overtwisted object called the overtwisted orange which is middledimensional and contractible.
We initiate a systematic study of convex hypersurface theory and generalize the bypass attachment to arbitrary dimensions. We also introduce a new type of overtwisted object called the overtwisted orange which is middledimensional and contractible.
(SarkarScadutoStoffregen) An odd Khovanov homotopy type
ArXiv
For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers OzsvathRasmussenSzabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of LawsonLipshitzSarkar of the even Khovanov stable homotopy type. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. We also construct a Z/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of X^j_o(L).
For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers OzsvathRasmussenSzabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of LawsonLipshitzSarkar of the even Khovanov stable homotopy type. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. We also construct a Z/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of X^j_o(L).
(BrownDeconinckDekimpeStaecker) Lifting classes for the fixed point theory of $n$valued maps
ArXiv
The theory of lifting classes and the Reidemeister number of singlevalued maps of a finite polyhedron $X$ is extended to $n$valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl. The liftings of an $n$valued map $f$ split into selfmaps of the universal covering space of $X$ that we call liftfactors. An equivalence relation is defined on the liftfactors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the liftfactors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$valued map equals its Reidemeister number. If an $n$valued map splits into $n$ singlevalued maps, then its $n$valued Reidemeister number is the sum of their Reidemeister numbers.
The theory of lifting classes and the Reidemeister number of singlevalued maps of a finite polyhedron $X$ is extended to $n$valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl. The liftings of an $n$valued map $f$ split into selfmaps of the universal covering space of $X$ that we call liftfactors. An equivalence relation is defined on the liftfactors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the liftfactors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$valued map equals its Reidemeister number. If an $n$valued map splits into $n$ singlevalued maps, then its $n$valued Reidemeister number is the sum of their Reidemeister numbers.
(AbelWillis) Colored KhovanovRozansky homology for infinite braids
ArXiv
We show that the limiting unicolored $\mathfrak{sl}(N)$ KhovanovRozansky chain complex of any infinite positive braid categorifies a highestweight projector. This result extends an earlier result of Cautis categorifying highestweight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLYPT KhovanovRozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLYPT homology of any braid positive link and the stable HOMFLYPT homology of the infinite torus knot as computed by Hogancamp.
We show that the limiting unicolored $\mathfrak{sl}(N)$ KhovanovRozansky chain complex of any infinite positive braid categorifies a highestweight projector. This result extends an earlier result of Cautis categorifying highestweight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLYPT KhovanovRozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLYPT homology of any braid positive link and the stable HOMFLYPT homology of the infinite torus knot as computed by Hogancamp.
(LipshitzSarkar) Spatial refinements and Khovanov homology
ArXiv
We review the construction and context of a stable homotopy refinement of Khovanov homology.
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.
(BehrensHillHopkinsMahowald) 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.
(LawsonLipshitzSarkar) 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.
(HendricksLipshitzSarkar) A simplicial construction of Gequivariant Floer homology
ArXiv
For G a Lie group acting on a symplectic manifold $(M,\omega)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a Gequivariant Floer cohomology of $L_0$ and $L_1$.
For G a Lie group acting on a symplectic manifold $(M,\omega)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a Gequivariant Floer cohomology of $L_0$ and $L_1$.
(BaoHonda) Semiglobal Kuranishi charts and the definition of contact homology
ArXiv
We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$, we define an algebra $HC(\mathcal{D})$. If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semiglobal Kuranishi charts.
We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$, we define an algebra $HC(\mathcal{D})$. If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semiglobal Kuranishi charts.