Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.
problem The study of rational homotopy groups of the space of long embeddings of codimension 2.
method Utilizing hairy graphs, the authors construct elements in the homotopy groups and prove their nontriviality.
result The rational homotopy groups of the space of long embeddings are infinite-dimensional in infinitely many degrees.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
problem Determining homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
method Computes rational homotopy groups of classifying spaces of diffeomorphisms.
result Determines rational pseudoisotopy stable range for compact spin manifolds.
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. 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.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
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 S1-equivaria…
The article proves estimates on homotopy and cohomology dimensions in fibrations.
problem Estimates on dimensions of homotopy and cohomology groups in fibrations.
method Proves estimates in formal elliptic spaces and specific cases.
result Proves estimates on dimensions of homotopy and cohomology groups in fibrations.
Given a finite metric CW complex X and an element α∈πn(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.
problem Challenges in rational homotopy theory for smooth spaces with arbitrary fundamental groups.
method Utilizes local systems over simplicial sets and a model structure for diffeological spaces.
result Establishes an equivalence between fibrewise rational diffeological spaces and algebraic local systems.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
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 B4(c)⊂R4 into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form Mλ=(S2×S2,μω0⊕ω0) where ω0…
New configuration space integrals show nontrivial formal smooth structures on 4-manifold bundles.
problem Disproving the 4-dimensional Smale conjecture by constructing nontrivial bundles.
method Defining new configuration space integrals relying on formal smooth structures.
result Discovering a generalized Miller-Morita-Mumford class obstructing formal smooth structures.
We study the rational homotopy of the moduli space NX of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g≥2. The symplectic group Aut(H1(X,Z))=Sp(2g,Z) has a natural action on the rational homotopy gr…
New invariant P helps classify simply-connected 8-manifolds.
problem Classifying simply-connected 8-manifolds.
method Introducing a rational homotopy invariant P and using it to classify manifolds.
result Simply-connected 8-manifolds are classified by the value of P.
New topological realization of Kontsevich graph complex for large dimensions.
problem Understanding the rational homotopy groups of Diff partial(D2k).
method Construction of a chain map from Kontsevich graph complex to rational singular chain complex.
result New elements in rational homotopy groups of BDiff partial(D2k) determined by cycles in graph complex.
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
Study on high-dimensional solid tori reveals infinite generation in their diffeomorphism groups.
problem Infinite generation in the homotopy groups of high-dimensional solid tori diffeomorphisms.
method Analysis of homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of the manifold to a form of Hermitian K-theory of the integral group ring of π1(S1).
result Homotopy groups of diffeomorphisms of high-dimensional solid tori are infinite in certain degrees.
This paper studies the rational homotopy groups of the group Diff(S4) of self-diffeomorphisms of S4 with the C∞-topology. We present a method to prove that there are many `exotic' non-trivial elements in π∗Diff(S4)⊗Q parametrized by trivalent graphs. As a corollary of…
Abstract: Rational decomposition of homeomorphism spaces for manifolds.
problem Constructing rational homotopy pullback decompositions for homeomorphism spaces.
method Rational homotopy pullback decomposition, nullhomotopy of stabilisation maps, tensor products of truncated operads.
result Rational section of the stabilisation map for homeomorphisms of R^d.
Scalable spaces are simply connected manifolds with nice cohomology properties.
problem Understanding the limitations of formality in higher homotopy groups.
method Analyzing the embedding of cohomology algebras into differential forms.
result Spaces that are formal but not scalable provide counterexamples to Gromov's conjecture.
The study determines fiber homotopy trivial bundles and their impact on curvature.
problem Understanding fiber homotopy trivial bundles and their effect on curvature.
method Classical approach via block bundles and surgery theory.
result Existence of elements of infinite order in homotopy groups of spaces of positive curvature.
New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.
problem Characterizing nonrigidity of open contractible manifolds.
method Construction of infinitely many pairwise nonhomeomorphic smooth open contractible manifolds.
result Existence of infinitely many pairwise nonhomeomorphic smooth open contractible 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.
problem Existence of elements in curvature spaces.
method Analyzes homotopy groups of spaces with positive curvature.
result Proves existence of infinite order elements in high-dimensional spaces with positive curvature.
Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
The paper provides a differential form interpretation of a theorem about the dimensions of rational homotopy groups of Diff(D^4).
problem Lower bounds of dimensions of rational homotopy groups of Diff(D^4) in terms of graph homology.
method Differential form interpretation and extension to arbitrary even dimensions.
result The proof can be extended to arbitrary even dimensions and made accessible to more readers.
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.
problem Lack of effective methods to connect geometric and homotopy information.
method Introducing Chen's iterated integrals on loop spaces.
result Upper bounds for Gromov's distortion and non-existence of small-volume cycles.
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ 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 S2×S2 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.
problem Realizing symplectic algebras and homotopy types on manifolds.
method Addressing questions on realizability of symplectic algebras and rational homotopy types by closed symplectic manifolds.
result Realization of symplectic algebras and homotopy types in various dimensions.
Maps between classifying spaces for certain groups are studied, with rational cohomology results.
problem Analyzing maps between classifying spaces for specific groups.
method Study of maps Θ between null-components of homomorphism and based map spaces. result Surjectivity of map Θ in rational cohomology for certain groups, and non-surjectivity for others. 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.
problem Computing the rational homotopy group of long embeddings.
method Invented two graph homologies and constructed a map between them.
result A monomorphism from top hairy graph homology to top BCR graph homology.
The study constructs bundles with non-multiplicative A-genus and finds non-trivial homotopy groups in spaces of metrics.
problem Locating non-trivial rational homotopy groups in spaces of metrics with lower curvature bounds.
method Constructs smooth bundles with non-vanishing A-genus and uses them to find homotopy groups.
result Non-trivial homotopy groups in spaces of metrics with lower curvature bounds.
Researchers determine the rational abelianization of a subgroup of mapping class groups.
problem Understanding the structure of the Chillingworth subgroup of mapping class groups.
method Using Johnson homomorphism and Casson-Morita homomorphism, they compute the abelianization and order of related Euler classes.
result They find the rational abelianization of the Chillingworth subgroup as a full mapping class group module.
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions.