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).
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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.
- Higher Dimensional Contact Topology (April 2024)
- Geometry and Topology Workshop (January 2024)
- Floer Homotopical Methods in Low Dimensional and Symplectic Topology (November 2022)
- Spectral Methods in Algebra, Geometry, and Topology (September 2022 - December 2022)
- Equivariant Techniques in Stable Homotopy (May 2021 / September 2022)
- 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.
(Chatterjee-Etnyre-Min-Mukherjee) Existence and construction of non-loose knots
ArXiv
In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling.
In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling.
(Yuan) A link invariant from higher-dimensional Heegaard Floer homology
ArXiv
We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p\colon W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We represent a link by a $\kappa$-strand braid, which is expressed as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $\kappa$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant. This work constitutes part of the author's PhD thesis.
We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p\colon W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We represent a link by a $\kappa$-strand braid, which is expressed as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $\kappa$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant. This work constitutes part of the author's PhD thesis.
(ReisinTzur) A Floer-theoretic interpretation of the polynomial representation of the double affine Hecke algebra
ArXiv
We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of $\kappa$-tuples of cotangent fibers and $\kappa$-tuples of conormal bundles of homotopically nontrivial simple closed curves in $T^*\Sigma$ with a certain braid skein group, where $\Sigma$ is a closed oriented surface of genus $> 0$ and $\kappa$ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to $\Sigma$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $\Sigma=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies.
We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of $\kappa$-tuples of cotangent fibers and $\kappa$-tuples of conormal bundles of homotopically nontrivial simple closed curves in $T^*\Sigma$ with a certain braid skein group, where $\Sigma$ is a closed oriented surface of genus $> 0$ and $\kappa$ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to $\Sigma$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $\Sigma=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies.
(Balmer-Ocal) Universal support for triangulated categories
ArXiv
We revisit a result of Gratz and Stevenson on the universal space that carries supports for objects of a triangulated category, in the absence of a tensor product.
We revisit a result of Gratz and Stevenson on the universal space that carries supports for objects of a triangulated category, in the absence of a tensor product.
(Ocal) Symmetric monoidal equivalences of quantum field theories in dimension two and Frobenius algebras
ArXiv
We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.
We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.
(Breen-Honda-Huang) The Giroux correspondence in arbitrary dimensions
ArXiv
We establish the Giroux correspondence in arbitrary dimensions. As corollaries we (i) give an alternate proof of a result of Giroux-Pardon that states that any Weinstein domain is Weinstein homotopic to one which admits a Weinstein Lefschetz fibration and (ii) prove that any two Weinstein Lefschetz fibrations whose Weinstein domain structures are Weinstein homotopic are related by the Weinstein Lefschetz fibration moves, affirming a conjecture of Giroux-Pardon.
We establish the Giroux correspondence in arbitrary dimensions. As corollaries we (i) give an alternate proof of a result of Giroux-Pardon that states that any Weinstein domain is Weinstein homotopic to one which admits a Weinstein Lefschetz fibration and (ii) prove that any two Weinstein Lefschetz fibrations whose Weinstein domain structures are Weinstein homotopic are related by the Weinstein Lefschetz fibration moves, affirming a conjecture of Giroux-Pardon.
(Popović) Link Floer homology splits into snake complexes and local systems
ArXiv
We classify isomorphism and chain homotopy equivalence classes of finitely generated $\mathbb{Z} \oplus \mathbb{Z}$ graded free chain complexes over $\frac{\mathbb{F}[U,V]}{(UV)}$. As a corollary, we establish that every link Floer complex $CFL(Y, L)$ over the ring $\frac{\mathbb{F}[U,V]}{(UV)}$ splits uniquely as a direct sum of snake complexes and local systems. This generalizes and extends the results of Petkova and Dai, Hom, Stoffregen, and Truong. We give the first example of an essentially infinite knot Floer-like complex, i.e., a complex satisfying all formal properties of link Floer complexes of knots in $S^3$ and whose chain homotopy equivalence class does not admit a representative of the form $C \otimes_{\mathbb{F}} \mathbb{F}[U,V]$. Finally, we also describe the first example of a knot Floer-like complex that does not admit a simultaneously vertically and horizontally simplified basis.
We classify isomorphism and chain homotopy equivalence classes of finitely generated $\mathbb{Z} \oplus \mathbb{Z}$ graded free chain complexes over $\frac{\mathbb{F}[U,V]}{(UV)}$. As a corollary, we establish that every link Floer complex $CFL(Y, L)$ over the ring $\frac{\mathbb{F}[U,V]}{(UV)}$ splits uniquely as a direct sum of snake complexes and local systems. This generalizes and extends the results of Petkova and Dai, Hom, Stoffregen, and Truong. We give the first example of an essentially infinite knot Floer-like complex, i.e., a complex satisfying all formal properties of link Floer complexes of knots in $S^3$ and whose chain homotopy equivalence class does not admit a representative of the form $C \otimes_{\mathbb{F}} \mathbb{F}[U,V]$. Finally, we also describe the first example of a knot Floer-like complex that does not admit a simultaneously vertically and horizontally simplified basis.
(Popović) Algebraic realizability of knot Floer-like complexes
ArXiv
We study algebraic obstructions to realizability of local equivalence classes of knot-like complexes. We classify local equivalence classes of knot-like complexes over $\mathbb{F}[U,V]$, answering a question of Dai, Hom, Stoffregen and Truong.
We study algebraic obstructions to realizability of local equivalence classes of knot-like complexes. We classify local equivalence classes of knot-like complexes over $\mathbb{F}[U,V]$, answering a question of Dai, Hom, Stoffregen and Truong.
(Etnyre-Min-Piccirillo-Roy) Small symplectic caps and embeddings of homology balls in the complex projective plane
ArXiv
We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in $\mathbb{C}\mathrm{P}^2$, and thereby provide the first examples of (infinitely many) symplectic handlebody decompositions of a closed symplectic 4-manifold. Our constructions provide a new topological interpretation of almost toric fibrations of $\mathbb{C}\mathrm{P}^2$ in terms of symplectic handlebody decompositions.
We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in $\mathbb{C}\mathrm{P}^2$, and thereby provide the first examples of (infinitely many) symplectic handlebody decompositions of a closed symplectic 4-manifold. Our constructions provide a new topological interpretation of almost toric fibrations of $\mathbb{C}\mathrm{P}^2$ in terms of symplectic handlebody decompositions.
(Chen) Deformations of objects in $n$-categories
ArXiv
In this paper, we prove that the deformation theory of an object in an $n$-category is controlled by the its $n$-fold endomorphism algebra. This recovers Lurie's results on deforming objects and categories. We also generalize a previous result by Blanc et al. on deforming a category and an object simultaneously to the case of $n$-categories.
In this paper, we prove that the deformation theory of an object in an $n$-category is controlled by the its $n$-fold endomorphism algebra. This recovers Lurie's results on deforming objects and categories. We also generalize a previous result by Blanc et al. on deforming a category and an object simultaneously to the case of $n$-categories.
(Blumberg-Hill-Mandell) Norms for compact Lie groups in equivariant stable homotopy theory
ArXiv
We propose a construction of an analogue of the Hill-Hopkins-Ravenel relative norm $N_{H}^{G}$ in the context of a positive dimensional compact Lie group $G$ and closed subgroup $H$. We explore expected properties of the construction. We show that in the case when $G$ is the circle group (the unit complex numbers), the proposed construction here agrees with the relative norm constructed by Angeltveit, Gerhardt, Lawson, and the authors using the cyclic bar construction. Our construction is based on a new perspective on equivariant factorization homology, using framings to convert from actions of one group to another.
We propose a construction of an analogue of the Hill-Hopkins-Ravenel relative norm $N_{H}^{G}$ in the context of a positive dimensional compact Lie group $G$ and closed subgroup $H$. We explore expected properties of the construction. We show that in the case when $G$ is the circle group (the unit complex numbers), the proposed construction here agrees with the relative norm constructed by Angeltveit, Gerhardt, Lawson, and the authors using the cyclic bar construction. Our construction is based on a new perspective on equivariant factorization homology, using framings to convert from actions of one group to another.
(Ballesteros-Gubitosi-GutierrezSagredo-Herranz) $κ$-Galilean and $κ$-Carrollian noncommutative spaces of worldlines
ArXiv
The noncommutative spacetimes associated to the $\kappa$-Poincar\'e relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $\kappa$-Galilei and $\kappa$-Carroll symmetries as contractions of the corresponding $\kappa$-Poincar\'e space and we show that these three spaces are defined by different algebras. In particular, the $\kappa$-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the $\kappa$-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.
The noncommutative spacetimes associated to the $\kappa$-Poincar\'e relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $\kappa$-Galilei and $\kappa$-Carroll symmetries as contractions of the corresponding $\kappa$-Poincar\'e space and we show that these three spaces are defined by different algebras. In particular, the $\kappa$-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the $\kappa$-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.
(Tian-Yuan) An example of higher-dimensional Heegaard Floer homology
ArXiv
We count pseudoholomorphic curves in the higher-dimensional Heegaard Floer homology of disjoint cotangent fibers of a two dimensional disk. We show that the resulting algebra is isomorphic to the Hecke algebra associated to the symmetric group.
We count pseudoholomorphic curves in the higher-dimensional Heegaard Floer homology of disjoint cotangent fibers of a two dimensional disk. We show that the resulting algebra is isomorphic to the Hecke algebra associated to the symmetric group.
(Horváth) Translation beyond Delsarte
ArXiv
We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets in certain $L^p_\mu$ spaces are given. Pego-type characterizations are also derived. Finally for some examples the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.
We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets in certain $L^p_\mu$ spaces are given. Pego-type characterizations are also derived. Finally for some examples the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.
(Krutowski) Combinatorial proof of Maslov index formula in Heegaard Floer Theory
ArXiv
We prove Lipshitz's Maslov index formula in Heegaard Floer homology via the combinatorics of Heegaard diagrams.
We prove Lipshitz's Maslov index formula in Heegaard Floer homology via the combinatorics of Heegaard diagrams.
(Beaudry-Hill-Lawson-Shi-Zeng) On the slice spectral sequence for quotients of norms of Real bordism
ArXiv
In this paper, we study equivariant quotients of the multiplicative norm $MU^{((C_{2^n}))}$ of the Real bordism spectrum by permutation summands, a concept defined here. These quotients are interesting because of their relationship to the so-called "higher real $K$-theories". We provide new tools for computing the equivariant homotopy groups of such quotients of $MU^{((C_{2^n}))}$ and quotients of the closely related spectrum $BP^{((C_{2^n}))}$. As a new example, we study spectra denoted by $BP^{((C_{2^n}))}\langle m,m\rangle$, which have non-trivial chromatic localizations only at heights equal to $rm$ where $0\leq r\leq 2^{n-1}$. These spectra are natural equivariant generalizations of integral Morava-$K$-theories. For $\sigma$ the real sign representation of $C_{2^{n-1}}$, we give a complete computation of the $a_{\sigma}$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$. We do this by establishing a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. We also give a complete computation of the $a_{\lambda}$-localized slice spectral sequence of $BP^{((C_{4}))}\langle 2,2\rangle$ for $\lambda$ a rotation of $\mathbb{R}^2$ by an angle of $\pi/2$. The non-localized slice spectral sequences can be recovered completely from these localizations.
In this paper, we study equivariant quotients of the multiplicative norm $MU^{((C_{2^n}))}$ of the Real bordism spectrum by permutation summands, a concept defined here. These quotients are interesting because of their relationship to the so-called "higher real $K$-theories". We provide new tools for computing the equivariant homotopy groups of such quotients of $MU^{((C_{2^n}))}$ and quotients of the closely related spectrum $BP^{((C_{2^n}))}$. As a new example, we study spectra denoted by $BP^{((C_{2^n}))}\langle m,m\rangle$, which have non-trivial chromatic localizations only at heights equal to $rm$ where $0\leq r\leq 2^{n-1}$. These spectra are natural equivariant generalizations of integral Morava-$K$-theories. For $\sigma$ the real sign representation of $C_{2^{n-1}}$, we give a complete computation of the $a_{\sigma}$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$. We do this by establishing a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. We also give a complete computation of the $a_{\lambda}$-localized slice spectral sequence of $BP^{((C_{4}))}\langle 2,2\rangle$ for $\lambda$ a rotation of $\mathbb{R}^2$ by an angle of $\pi/2$. The non-localized slice spectral sequences can be recovered completely from these localizations.
(Lipshitz-Sarkar) Khovanov homology of strongly invertible knots and their quotients
ArXiv
We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology distinguishes certain slice disks. We also give an analogous spectral sequence for the Heegaard Floer homology of the branched double cover.
We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology distinguishes certain slice disks. We also give an analogous spectral sequence for the Heegaard Floer homology of the branched double cover.
(Cheng-Hedden-Sarkar) Murasugi sum and extremal knot Floer homology
ArXiv
The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $\tau=g$ for each summand if and only if $\tau=g$ holds for the Murasugi sum (with $\tau$ and $g$ defined appropriately for multi-component links). Some applications are presented.
The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $\tau=g$ for each summand if and only if $\tau=g$ holds for the Murasugi sum (with $\tau$ and $g$ defined appropriately for multi-component links). Some applications are presented.
(Honda-Tian-Yuan) Higher-dimensional Heegaard Floer homology and Hecke algebras
ArXiv
Given a closed oriented surface $\Sigma$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$ and show that $\mathcal{F}$ is an isomorphism of algebras. We also establish analogous results for punctured surfaces.
Given a closed oriented surface $\Sigma$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$ and show that $\mathcal{F}$ is an isomorphism of algebras. We also establish analogous results for punctured surfaces.
(AngeliniKnoll-Gerhardt-Hill) 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 anti-involution, 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 anti-involution. 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 anti-involution, 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 anti-involution. 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.
(Hill-Lawson) $E_k$-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.
(Lipshitz-Sarkar) A mixed invariant of non-orientable surfaces in equivariant Khovanov homology
ArXiv
We construct a mixed invariant of non-orientable surfaces from the Lee and Bar-Natan 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 non-orientable surfaces from the Lee and Bar-Natan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including some exotic examples.
(Breen-Christian) Mitsumatsu's Liouville domains are stably Weinstein
ArXiv
We develop explicit local operations that may be applied to Liouville domains, with the goal of simplifying the dynamics of the Liouville vector field. These local operations, which are Liouville homotopies, are inspired by the techniques used by Honda and Huang in [HH19] to show that convex hypersurfaces are $C^0$-generic in contact manifolds. As an application, we use our operations to show that Mitsumatsu's well-known Liouville-but-not-Weinstein domains are stably Weinstein, answering a question asked by Huang in [Hua20].
We develop explicit local operations that may be applied to Liouville domains, with the goal of simplifying the dynamics of the Liouville vector field. These local operations, which are Liouville homotopies, are inspired by the techniques used by Honda and Huang in [HH19] to show that convex hypersurfaces are $C^0$-generic in contact manifolds. As an application, we use our operations to show that Mitsumatsu's well-known Liouville-but-not-Weinstein domains are stably Weinstein, answering a question asked by Huang in [Hua20].
(Manolescu-Sarkar) 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 pseudo-holomorphic 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 pseudo-holomorphic 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.
(Hill-Mehrle-Quigley) 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.
(Lawson-Lipshitz-Sarkar) 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.
(Blumberg-Hill) Bi-incomplete 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 bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete 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 bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete 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.
(Beaudry-Hill-Lawson-Shi-Zeng) 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 finite-dimensional 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 finite-dimensional 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.
(Akhmechet-Krushkal-Willis) 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.
(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 show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle 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 3-manifolds. Among other results, we complete the classification up to orientation-preserving 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 2-handle 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.
(Colin-Honda-Tian) Applications of higher-dimensional Heegaard Floer homology to contact topology
ArXiv
The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.
The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.
(Beaudry-Hill-Shi-Zeng) 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 "red-shifting" 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 non-trivial 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 "red-shifting" 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 non-trivial 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.
(Ghiggini-Colin-Honda) An exposition of the equivalence of Heegaard Floer homology and embedded contact homology
ArXiv
This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in arXiv:1208.1074, arXiv:1208.1077 and arXiv:1208.1526.
This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in arXiv:1208.1074, arXiv:1208.1077 and arXiv:1208.1526.
(Beaudry-Hill-Shi-Zeng) Models of Lubin-Tate 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 Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ 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 Hill-Hopkins-Ravenel 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 Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ 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 Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.
(Akhmechet-Krushkal-Willis) 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.
(Breen) Morse-Smale characteristic foliations and convexity in contact manifolds
ArXiv
We generalize a result of Giroux which says that a closed surface in a contact $3$-manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori is $C^{\infty}$-close to a convex hypersurface.
We generalize a result of Giroux which says that a closed surface in a contact $3$-manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori is $C^{\infty}$-close to a convex hypersurface.
(Manolescu-Marengon-Sarkar-Willis) A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds
ArXiv
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous 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 Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B^2 \times S^2$, $S^1 \times B^3$, $\mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous 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 Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B^2 \times S^2$, $S^1 \times B^3$, $\mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.
(Lipshitz-Sarkar) Khovanov homology detects split links
ArXiv
Extending ideas of Hedden-Ni, 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 well-defined up to quasi-isomorphism.
Extending ideas of Hedden-Ni, 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 well-defined up to quasi-isomorphism.
(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 Hill--Hopkins--Ravenel. 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 co-Tambara 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 Hill--Hopkins--Ravenel. 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 co-Tambara 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 $BBU_{\mathbb R}$.
(Lawson-Lipshitz-Sarkar) Chen-Khovanov spectra for tangles
ArXiv
We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of Chen-Khovanov 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 Chen-Khovanov 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 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
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. 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 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.
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. 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 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.
(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. 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 JSJ-type 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 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 $\mathrm{Tait}(K)$ 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 $\dim J^\flat(W_1) = 58$.
(Honda-Huang) 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 h-principle 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 h-principle for codimension 2 contact submanifolds.
(Willis) Khovanov-Rozansky homology for infinite multi-colored braids
ArXiv
We define a limiting $\mathfrak{sl}_N$ 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. 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 semi-infinite and bi-infinite multi-colored braids.
We define a limiting $\mathfrak{sl}_N$ 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. 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 semi-infinite and bi-infinite multi-colored 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 X-action 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 X-action on Lee's perturbation of Khovanov homology.
(Lee) Towards a higher-dimensional construction of stable/unstable Lagrangian laminations
ArXiv
We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov 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 higher-dimensional 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 higher-dimensional generalization of a geodesic lamination.
We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov 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 higher-dimensional 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 higher-dimensional 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 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 define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'ath-Szab\'o's gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.
We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'ath-Szab\'o's gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.
(Manion-Marengon-Willis) Generators, relations, and homology for Ozsváth-Szabó's Kauffman-states algebras
ArXiv
We give a generators-and-relations 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 generators-and-relations 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.
(Beaudry-Bobkova-Hill-Stojanoska) 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 $\#^r(S^2\times S^1)$
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 null-homologous 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 handle-slides, 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 null-homologous 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 handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.
(Colin-Honda) 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 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. The proof of the second statement uses the Handel-Miller theory of end-periodic diffeomorphisms of end-periodic 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 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. The proof of the second statement uses the Handel-Miller theory of end-periodic diffeomorphisms of end-periodic surfaces.
(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 $BP^{((C_4))}\langle 2 \rangle$. 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 $E_4^{hC_{12}}$ is 384-periodic.
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 height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ 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
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).
(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.
(Christian-Menke) A JSJ-type decomposition theorem for symplectic fillings
ArXiv
We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if $(M,\xi)$ is obtained from $(S^3,\xi_{\mathrm{std}})$ via Legendrian surgery along a knot which has been stabilized both positively and negatively, then $(M,\xi)$ has a unique exact filling.
We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if $(M,\xi)$ is obtained from $(S^3,\xi_{\mathrm{std}})$ via Legendrian surgery along a knot which has been stabilized both positively and negatively, then $(M,\xi)$ has a unique exact filling.
(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 $S^3$.
(Honda-Huang) Bypass attachments in higher-dimensional 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 middle-dimensional 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 middle-dimensional and contractible.
(Sarkar-Scaduto-Stoffregen) 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 Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar 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 Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar 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).
(Brown-Deconinck-Dekimpe-Staecker) Lifting classes for the fixed point theory of $n$-valued maps
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 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 self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors 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 lift-factors 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 $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.
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 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 self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors 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 lift-factors 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 $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.
(Abel-Willis) Colored Khovanov-Rozansky homology for infinite braids
ArXiv
We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. 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.
We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. 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
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.
(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 $(M,\omega)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a G-equivariant 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 G-equivariant Floer cohomology of $L_0$ and $L_1$.
(Bao-Honda) Semi-global 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 semi-global 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 semi-global Kuranishi charts.