Characterizes primary operations in differential cohomology using stacks.
problem Understanding primary operations in differential cohomology.
method Characterization via stacks, explicit refinement of Steenrod squares and powers, interplay between different cohomology types.
result Developed techniques for differential cohomology, including Künneth decomposition.
Combining classical and modern topology, shows maps between certain manifolds must have degree zero.
problem Degree zero maps between certain manifolds
method Combining Thom's work on the Steenrod problem with simplicial volume, using Thom spaces and Steenrod powers
result Every map between certain manifolds must have degree zero
Proposes a method to compute the second Steenrod square for odd Khovanov homology.
problem Computing the second Steenrod square for odd Khovanov homology.
method Proposes a new method to compute the second Steenrod square, showing it to be a link invariant.
result Shows the proposed method gives a refinement of the Rasmussen s-invariant with Z/2Z coefficients. New knots found with non-trivial Steenrod operations on Khovanov homology.
problem Identifying knots with non-trivial Steenrod operations on Khovanov homology.
method Examined prime, hyperbolic, and satellite knots using Steenrod operations.
result Found knots (prime, hyperbolic, satellite) with non-trivial Steenrod operations on Khovanov homology.
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
Proves a local version of Myers-Steenrod theorem for specific manifolds.
problem Generalization of Myers-Steenrod theorem to local topological groups.
method Proof for local topological groups of isometries acting on specific manifolds.
result New regularity result for locally homogeneous Riemannian metrics.
Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.
problem Computing Steenrod squares on Khovanov homology.
method Spatial refinements of even and odd Khovanov homology, computation of Sq^2.
result Steenrod squares Sq^2 on Khovanov homology spaces are determined for knots up to 11 crossings.
If F is a family of mod 2 flat k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.
This paper uses Steenrod homology to simplify surgery on generalized manifolds.
problem Simplifying the surgery process on generalized manifolds.
method Develops Steenrod homology theory to avoid geometric splitting in surgery.
result Constructs elements of Steenrod homology groups for generalized manifolds.
New operations match Steenrod squares on Khovanov homology.
problem Matching Steenrod squares with operations on Khovanov homology.
method Applied cup-i products to Khovanov functor and proved agreement with Steenrod squares.
result Lipshitz-Sarkar's Sq^2 agrees with Morán's sq^2.
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
We compute Steenrod squares on Khovanov homology.
problem Computing Steenrod squares on Khovanov homology.
method Stable cup-i products on cochain complexes of augmented semi-simplicial objects in the Burnside category.
result Explicit formulas for cohomology operations on Khovanov homology.
New algorithm calculates Steenrod squares in Khovanov cohomology.
problem Computing Steenrod squares in Khovanov cohomology.
method Flow category simplification techniques to calculate second Steenrod square and Bockstein homomorphisms.
result Observation of new homotopy types and evidence against CP2 summands. We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a c…
The Myers-Steenrod theorem is extended to Finsler manifolds with low regularity.
problem Extending the Myers-Steenrod theorem to Finsler manifolds with minimal regularity.
method Reduction of Finslerian problems to Riemannian ones using the Binet-Legendre metric.
result Isometries between Ck,α-smooth Finsler metrics are diffeomorphisms of class Ck+1,α. Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…
Computes cohomology of Steenrod algebra for k ≤ 5.
problem Determining a basis of cohomology for Steenrod algebra.
method Algorithm based on generators and Adams relations.
result Verification of hand-computed results for k = 4.
Defines symplectic homology for Liouville cobordisms, satisfying Eilenberg-Steenrod axioms.
problem Defining and satisfying Eilenberg-Steenrod axioms for symplectic homology.
method Definition and proof of axioms for symplectic homology of Liouville cobordisms.
result Symplectic homology satisfies Eilenberg-Steenrod axioms except for dimension.
Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khova…
The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A …
In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, …
In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…
New proofs confirm travel time data determine simple metrics on a disc.
problem Determining a simple Riemannian metric from travel time data.
method Proofs based on Myers-Steenrod theorem, Lipschitz-type stability estimate.
result Travel time data determine a simple Riemannian metric on a disc up to natural gauge.
Develops Floer theory for 3-manifold covers using equivariant structures.
problem Calculating Steenrod operations and Heegaard Floer homology of 3-manifold covers.
method Equivariant Floer theory with Z/2-actions and localization. result Localization to twisted Floer cohomology for invariant sets.
Harmonic coordinates for Finsler manifolds prove a theorem but not optimal regularity.
problem Proving the Myers--Steenrod theorem for Finsler manifolds.
method Existence of harmonic coordinates for nonlinear Finsler Laplacian.
result Partial results on optimal regularity for Berwald metrics.
New homotopy types defined for links in thickened surfaces with higher genus.
problem Defining stable homotopy types for links in surfaces with higher genus.
method Defined Khovanov-Lipshitz-Sarkar homotopy types and Steenrod squares for links in thickened surfaces with genus > 1.
result First meaningful Khovanov-Lipshitz-Sarkar stable homotopy types for links in 3-manifolds other than the 3-sphere.
Spaces containing compact subsets with polyhedral complements are studied.
problem Characterizing and understanding spaces with specific topological properties.
method Introduced coronated polyhedra and used them to derive new cohomology and homotopy sequences.
result Spaces with the specified property have well-defined cohomology and homotopy sequences.
Introduces fine shape theory to simplify shape and antishape invariants.
problem Complexity and limitations of existing shape theories for metrizable spaces.
method Develops fine shape theory with a simple definition, aiming to supersede known shape theories.
result Fine shape theory unifies Čech cohomology and Steenrod-Sitnikov homology as invariants.
Study homotopy type of periodic links using Khovanov spectrum.
problem Understanding the homotopy type of periodic links.
method Construct Khovanov spectrum and analyze its cohomology.
result Khovanov spectrum admits a homology group action and Steenrod algebra structure.
Researchers compute degrees of maps between SU(2)-bundles over 5-sphere.
problem Computing degrees of maps between SU(2)-bundles over the 5-sphere.
method Using Steenrod squares to identify obstructions and constructing explicit maps.
result Explicit construction of maps realizing each integer degree.
Incompatible operations affect Khovanov homology and spectral sequences.
problem Incompatibility between operations and spectral sequences.
method Observation of obstructions to integral lifting and spectrification.
result Lipshitz-Sarkar Steenrod operations are incompatible with Szabo's spectral sequence.
Lifts an sl2 action to annular Khovanov homology's stable refinement.
problem Stable refinement of annular Khovanov homology's sl2 action. method Lifts actions of sl2 generators to maps of spectra, using cancellations in cube of resolutions. result Commutativity of sl2 action with Steenrod algebra action. This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the followin…
Affirmative answer to a question about a map extending normal invariants.
problem Whether a map extending normal invariants is bijective.
method Construction of a map extending normal invariants for arbitrary manifolds.
result The map is bijective for arbitrary manifolds.
This paper considers fundamental issues related to Finslerian isometries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Using distance coordinates, a simple proof is given for the Finslerian version of the Myers-Steenrod theorem and for t…
The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new to…
In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers-Steenrod's theorem for orbit spaces. These results are proved in the more general context of s…
The paper explores homological dimensions and dimensional full-valuedness in metric compacta.
problem Investigating homological dimensions and dimensional full-valuedness in metric compacta.
method Comparing Čech and Steenrod homology dimensions, proving dimensional full-valuedness for specific conditions.
result Two-dimensional metric compacta with specific properties satisfy dimensionally full-valuedness.
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …
Extends Adams' theorem to periodic cohomology.
problem Proving Adams' theorem for periodic cohomology.
method Adapting Adams' approach to periodic cohomology.
result Conjecture proven in a special case.
New examples show immersions not homologous to embeddings.
problem Tackles the question of immersions vs. embeddings in manifolds.
method Provides examples showing immersions not homologous to embeddings.
result Double points represent non-trivial homology classes.
We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real …
This is a survey paper, starting from the general notion of coordinate bundle taken from Steenrod. Its aim is to provide a motivation for the introduction of cyclic homology (and the closely related noncommutative de Rham cohomology) by Connes, Tsygan and the author. The bridge is made through a generalization of Chern…
Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.
problem Classifying Spin(7) structures on compact 8-manifolds.
method Obstruction theory applied to Spin(7) structures on compact 8-manifolds with abelian fundamental group.
result Compact Riemannian 8-manifolds with holonomy Spin(7) have exactly two Spin(7) structures extending the induced G2 structure on the boundary.
Reconstruct cohomology and homology from compact subsets and nerves.
problem Reconstructing cohomology and homology from compact subsets and nerves.
method Using Bousfield-Kan/Araki-Yoshimura type spectral sequences and corrected derived limits.
result Corrected derived limits coincide with usual ones when topology is discrete.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and exte…