arXiv research

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…

Study rational homotopy types of embedding spaces of manifolds.

Study shows equivariant Khovanov homotopy types are equivalent.

Defines homotopy type for links in thickened surfaces.

4-manifolds with specific groups have unique homotopy types.

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the h…

Study determines homotopy types of specific 6-manifolds.

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…

Paper calculates homotopy types of non-compact surfaces using groupoids.

Homotopy types of 4-manifolds tied to their fundamental groups.

The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.

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

Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.

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

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when the later are given their natural CW-decomposition with unique cells of order 0, $(n-1)$ and $n$; a result resembling J.H.C. Whi…

New examples of manifolds that are homotopy but not simple homotopy equivalent.

Proves PL cobordism category's homotopy type, analogous to smooth case.

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy ty…

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…

We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R^4 is not determined by the cohomology ring, thereby …

Homotopy types of curve and arc complexes are studied.

Study homotopy types of 4-manifolds, finding decompositions and conditions for desuspension.

Computes homotopy types of embedding spaces in tight contact 3-manifolds.

Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.

We discuss a question by Felix, Oprea, and Tanre concerning nonnegative curvature and (rational) homotopy type.

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…

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of…

Study realizes symplectic algebras and homotopy types on manifolds.

New examples of manifolds with similar homotopy but different simple homotopy types.

Study the space of embeddings of split links in 3D and 4D.

New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.

Satellite formula connects knot concordance invariants to surgery.

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves w…

Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.

Finite type and finitely generated homotopy groups for manifold automorphisms.

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

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is n…

Paper studies spaces of flattenings of simplicial spheres and their homotopy type.

Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.

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…

We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ colored links and quantum spin networks and derive some of its basic properties. In the case of $n$-colored B-adequate links, we show a stabilization of the homotopy types as the coloring $n\rightarrow\infty$, generalizing the tail behavior of the colored Jones …

Refines Khovanov homology using signed Burnside categories.

Proves a connectivity conjecture for free groups, showing homotopy type of spheres.