arXiv research

Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, the…

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 Kh…

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…

Refines Khovanov homology using signed Burnside categories.

Notes on Khovanov and knot Floer theories' stable homotopy types.

New homotopy theory reveals the structure of stable curves.

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $gl_n(S)$-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…

Defines homotopy type for links in thickened surfaces.

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

New homotopy types defined for links in thickened surfaces with higher genus.

Generalizes Floer homotopy via Morse-Bott theory.

We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for …

We will define a version of Seiberg-Witten-Floer stable homotopy types for a closed, oriented 3-manifold $Y$ with $b_1(Y) > 0$ and a spin-c structure $\mathfrak{c}$ on $Y$ with $c_1(\mathfrak{c})$ torsion under an assumption on $Y$. Using the Seiberg-Witten-Floer stable homotopy type, we will construct a gluing formula…

The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The proc…

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

Study of embedding spaces using homotopy theory and operads.

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…

The paper constructs multiple manifolds with similar properties.

New invariant for 4-manifolds with framed links, stronger than existing invariants.

This article constructs the moduli stack of torsionfree $G$-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any $\infty$-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.

The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the autho…

Study the topology of stable vector fields and Lyapunov functions on R^n.

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real close…

Computes homology of an obstruction chain complex in grid homology.

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …

Spatial refinement of Bar-Natan homology constructed.

Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.

Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. Th…

New infinite family of 4-manifolds with same stable properties but not homotopy equivalent.

The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group o…

Revisits Pontryagin's proof of stable stems 0, 1, and 2.

Researchers find a Steenrod square for link Floer homology.

This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.

In this paper, we study Conley theory in Hilbert spaces and make some refinement of the construction of the stable Conley index developed by Gȩba, Izydorek, and Pruszko. For instance, we allow subspaces other than invariant subspaces in the the construction. As a main result, we show that the resulting stable Conley in…

We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.

New stable homotopy refinement of quantum annular Khovanov homology.