arXiv research

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

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

Study homotopy groups of open books and their pages, pages, and bindings.

Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.

Novikov theorem extended to rational Pontryagin classes for cyclic group $C_4$.

We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply-connected four manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.

Characterizes Stein surfaces with finite homotopy rank-sum.

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivaria…

The article proves estimates on homotopy and cohomology dimensions in fibrations.

Given a finite metric CW complex $X$ and an element $α\in π_n(X)$, what are the properties of a geometrically optimal representative of $α$? We study the optimal volume of $kα$ as a function of $k$. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an…

We give the first explicit computations of rational homotopy groups of spaces of "long knots" in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E^1 term is defined in terms of braid Lie algebras. For odd k we establish a vanishing line for this spectral sequence,…

Introduces a framework for rational homotopy theory in diffeological spaces.

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…

We explore a relationship between topological properties of orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that evaluation map vanishies on the second homotopy group and obtain a Gottlieb-type vanishing theore…

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_λ= (S^2 \times S^2, μω_0 \oplus ω_0)$ where $ω_0$…

New configuration space integrals show nontrivial formal smooth structures on 4-manifold bundles.

We study the rational homotopy of the moduli space ${\mathcal N}_X$ of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface $X$ of genus $g\geq 2$. The symplectic group $Aut(H_1(X,{\mathbb Z}))=Sp(2g,{\mathbb Z})$ has a natural action on the rational homotopy gr…

New invariant P helps classify simply-connected 8-manifolds.

New topological realization of Kontsevich graph complex for large dimensions.

Study rational homotopy types of embedding spaces of manifolds.

Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.

Study on high-dimensional solid tori reveals infinite generation in their diffeomorphism groups.

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…

Abstract: Rational decomposition of homeomorphism spaces for manifolds.

The study determines fiber homotopy trivial bundles and their impact on curvature.

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

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in th…

Proves existence of infinite order elements in curvature spaces.

Study of symplectomorphisms on ruled surfaces under circle actions.

The paper provides a differential form interpretation of a theorem about the dimensions of rational homotopy groups of Diff(D^4).

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rationa…

For an orbifold M we define a homology group, called t-singular homology group t-H_q(M), which depends not only on the topological structure of the underlying space of M, but also on the orbifold structure of M. We prove that it is a b-homotopy invariant of orbifolds. If M is a manifold, t-H_q(M) coincides with the usu…

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

New method uses iterated integrals to bridge geometric and homotopy information.

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends…

The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the ra…

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundl…

Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as …

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…

Study realizes symplectic algebras and homotopy types on manifolds.

Maps between classifying spaces for certain groups are studied, with rational cohomology results.

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.

Two graph homologies help compute embedding space.

The study constructs bundles with non-multiplicative A-genus and finds non-trivial homotopy groups in spaces of metrics.

\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality.…

Researchers determine the rational abelianization of a subgroup of mapping class groups.

Study shows conditions for rational ellipticity of manifolds with symmetries.