arXiv research

Surveying ${\mathbb A}^1$-homotopy theory and contractible varieties.

The study confirms that certain symmetric spaces are formal.

The paper determines modular cohomotopy groups up to extensions using classical and unstable homotopy methods.

Refines quantum annular homology using stable homotopy methods.

The paper classifies bundles over complex projective plane.

Floer homotopy theory applies to Lagrangians, overcoming curvature issues.

Quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

Localizes smooth spaces to study their homotopy properties.

Paper introduces a new method for controlled surgery obstructions.

New category theory for complex projective plane sections.

Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.

Study extends gauge-theoretic invariant to higher-dimensional Kahler surfaces and calculates homotopy groups.

The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy c…

The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.

Paper extends gauge theory results to homology tori, preserving spin structure obstructions.

We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techni…

The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.

Almost complex structures found on many homotopy complex projective spaces.

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.

Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.

Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ o…

This thesis generalizes structures on $\mathcal{Q}$-manifolds and Lie $n$-algebroids.

An involutive distribution $C$ on a smooth manifold $M$ is a Lie-algebroid acting on sections of the normal bundle $TM/C$. It is known that the Chevalley-Eilenberg complex associated to this representation of $C$ possesses the structure $\mathbb{X}$ of a strong homotopy Lie-Rinehart algebra. It is natural to interpret …

We present homotopy theoretic and geometric interpretations of the Kane-Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the…

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks…

Geometrically counts embedded circles and sections of vector bundles.

Study the embedding space of a Hopf link in 3D and 3-manifolds.

New minimal surfaces in spheres with complex topologies from capillarity.

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial knots in $\mathbb{R}^3$ with the inductive limit topology coming from the spaces $\m…

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…

We show that the Hopf elements, the Kervaire classes, and the $\barκ$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-…

Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

This paper uses sheaf theory to constrain knot types in clean intersections.

The paper classifies 4-manifolds with given boundaries.

New invariants define the rational and real homotopy types of closed manifolds.

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

New formulas classify higher-dimensional knots and links.

This paper studies the rational homotopy groups of the group $\mathrm{Diff}(S^4)$ of self-diffeomorphisms of $S^4$ with the $C^\infty$-topology. We present a method to prove that there are many `exotic' non-trivial elements in $π_*\mathrm{Diff}(S^4)\otimes \mathbb{Q}$ parametrized by trivalent graphs. As a corollary of…

A new spin structure is constructed for a bundle of harmonic forms.

Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.

We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class i…

Study shows weak homotopy equivalences for complete minimal surfaces.

The paper resolves kinks on curves on surfaces with punctures.

Generalizes van Est map to geometric stacks and homotopy theory.

We investigate a relationship between Ozsváth and Szabó's bordered theory and the algebras and bimodules constructed by Khovanov-Seidel. Specifically, we show that (a variant of) a special case of Ozsváth-Szabó's algebras has a quotient which is isomorphic to the Khovanov-Seidel quiver algebra with coefficients in $\ma…