Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.
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48 results for “motivic homotopy”
The quadratic linking degreemath.AG
Introduces quadratic linking degree in algebraic geometry.
problem No specific problem stated; focuses on new concept.
method Motivic homotopy theory and Witt group.
result Explicit computation of quadratic linking degree.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
Geometric models for algebraic suspensions using affine deformation spaces.
problem Analyzing which motivic homotopy types have smooth scheme representatives.
method Affine deformation spaces of embeddings provide models for -suspensions of schemes.
result Yields many examples of -connected and strictly quasi-affine schemes.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version …
Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomol- ogy. Furthermore, we observe that the …
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.
Consider a closed non-degenerate 3-form with an infinitesimal action of a Lie algebra . Motivated by the fact that the observables associated to form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra . We formulate exist…
Banyaga has shown that the group of symplectomorphisms Symp(N) of a compact symplectic manifold (N,w) determines the symplectic structure. This motivates the study of the homotopy properties of Symp(N). Gromov has shown that the group of symplectomorphisms of N is homotopic to SO(3)\times SO(3) when N is the product of…
Develops derived differential geometry theory.
problem Homotopy and intersection in smooth manifolds.
method Using -algebras and homotopy transfer.
result Derived manifolds form a category of fibrant objects.
Study on descent properties of complex affine surfaces under proper morphisms.
problem Understanding descent behavior of homotopy-theoretic properties of smooth affine surfaces.
method Examined Eilenberg-MacLane property and introduced finite homotopy rank-sum property. Proved descent under proper morphisms for surfaces of log Kodaira dimension ≤0.
result Finite homotopy rank-sum property descends under proper morphisms for smooth affine surfaces of log Kodaira dimension ≤0.
Motivated by the definition of homotopy spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a -category with invertible -morphisms, and that certain fiber product property holds in this -category. In a subsequent pa…
Browder-Novikov-Sullivan-Wall surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory (such as the Wall surgery obstruction groups), without losing sight of the geometric…
New method recovers differential cohomology from diffeological spaces.
problem Recovering differential cohomology from diffeological spaces.
method Introducing skeletal diffeologies and thin homotopies to recover differential cohomology.
result Ordinary differential cohomology can be recovered in terms of the homotopy theory of skeletal diffeological spaces.
Defines a new invariant for 4-manifolds with boundary.
problem Homotopy classification of 4-manifolds with boundary.
method Definition of a relative k-invariant for CW pairs.
result Relative k-invariant is an obstruction to extending maps.
Maps of degree 1 and critical points on manifolds are studied.
problem Evaluate the minimal number of critical points on closed smooth manifolds.
method Examined maps of degree 1 and considered high-dimensional examples.
result In dimension 3 or less, for maps of degree 1.
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept o…
We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney…
It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is al…
Localizes smooth spaces to study their homotopy properties.
problem Understanding the homotopy theory of smooth spaces.
method Model category localization, Quillen equivalences, fibrant replacement.
result Localisation of smooth spaces agrees with motivic-style -localisation.
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved -algebras.
result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
Higher prequantum geometrymath-ph
This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey ex…
Study of point-pushing actions on manifolds with boundary.
problem Analyzing point-pushing actions on manifolds with boundary.
method Examining the homomorphism from braid groups to homotopy classes of homotopy equivalences.
result Injectivity of the map from the k-th braid group to homotopy classes of homotopy equivalences.
Constrained least squares regression is an essential tool for high-dimensional data analysis. Given a partition of input variables, this paper considers a particular class of nonconvex constraint functions that encourage the linear model to select a small number of variables from a small number of groups …
This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.
problem Extending classical contact structures to differentiable stacks.
method Introducing and -shifted contact structures on Lie groupoids, using line bundle-valued 1-forms and homotopy kernels.
result Definition and examples of and -shifted contact structures on Lie groupoids.
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
problem Formalizing and simplifying the study of partial differential equations (PDEs).
method Using cohomology of diffieties to formalize PDEs and their properties.
result Differential calculus on PDE solution spaces is homotopy calculus on horizontal De Rham algebras of diffieties.
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete examples such as wheeled completion of the operad of strongly homotopy Lie alge…
In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of di…
Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…
We describe two constructions giving rise to curved -algebras. The first consists of deforming -algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a …
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…
The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible mot…
This paper defines RII number for knot projections and shows it can be any nonnegative number.
problem Defining and quantifying the minimum number of specific types of deformations for knot projections.
method Using deformations of types 1, 2, and 3, analogs of Reidemeister moves, to simplify knot projections and define RII number.
result RII number can be any nonnegative number, not just zero as previously conjectured.
Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a…
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…
The Farey tree helps embed rational balls and lens spaces into complex projective space.
problem Embedding rational homology balls and lens spaces into complex projective space.
method Recursive Kirby calculus argument using the Farey tree.
result Explicit constructions of embeddings of triples of rational homology balls into homotopy .
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
On homotopy braidsmath.GR
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
Let L be a Legendrian knot in R^3 with the standard contact structure. In [10], a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreove…
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
New insights into Khovanov homology complexity and topological structure.
problem Complexity of computing Khovanov homology for closed braids.
method Analysis of independence simplicial complexes and polynomial time algorithms.
result Independence simplicial complexes associated to 4-braid diagrams are homotopy equivalent to wedges of spheres.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.