The paper proves topological finiteness for surfaces with finite Willmore energy.
problem Understanding the topology of surfaces with finite Willmore energy.
method Combining Allard regularity theorem and Reifenberg's topological disk theorem.
result Topological finiteness for a class of properly immersed surfaces with finite Willmore energy.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
The abstract discusses topological censorship with five hypotheses.
problem Proving topological censorship in spacetimes.
method Presented five hypotheses to prove a theorem.
result Proof of a topological censorship theorem.
Theorem proves topological censorship for universes with positive cosmological constant.
problem Proving topological censorship for spacetimes with positive cosmological constant.
method Developed a new theorem assuming eventual isolation of black hole collections.
result Regions near black hole collections have trivial fundamental group.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is (a)-regular. Here…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
Paper introduces topological eigenvalue theorems for tensor analysis in multi-modal data.
problem Lack of deep understanding of tensor structures in multi-modal data fusion.
method Introduces topological perspective to tensor eigenvalue analysis, linking eigenvalues to topological features.
result Establishes new theorems that enhance understanding of tensor structures in data fusion.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
Generalizes theorem for topological G-manifolds with linear Lie groups G.
problem Understanding topological G-manifolds with linear Lie groups G. method Using countable CW complexes and Palais-proper actions.
result Topological G-manifolds have G-homotopy type of countable G-CW complexes. In this paper we show that topological subgroupoids of Lie groupoids, under special circumstances are Lie subgroupoids. Giving an example, we indicate that having the same topological dimension is a necessary condition for topological subgroupoids to be Lie subgroupoids. Also, we provide some conditions for double subg…
Survey of topological 4-manifold theory, highlighting foundational theorems and pitfalls.
problem Understanding the topological properties of 4-dimensional manifolds.
method Compilation and explanation of foundational theorems, with cautionary notes.
result Many intuitive results in differential topology are not true in the topological category.
Revisit Fenn's table theorem from a differential-topological perspective.
problem Prove zero-existence theorem on a cylinder and horizontal square-table theorem under Fenn's boundary conditions.
method Differential-topological approach.
result Prove horizontal square-table theorem under more general boundary conditions.
Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.
Proofs a theorem using basic geometric tools.
problem C2-rectifiability problem
method Elementary geometric measure theory and topology
result Gives a proof of Alberti's Luzin-type theorem
Max flow/min cut theorem extended to currents and topology.
problem Continuous max flow/min cut theorem for complex domains.
method Continuous analogue of max flow/min cut theorem considering topology.
result Continuous max flow/min cut theorem proven for currents and laminations.
New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
The paper studies topological properties of Ricci shrinkers using weighted L2 cohomology.
problem Proving topological results for smooth gradient Ricci shrinkers.
method Weighted L2 cohomology and extensions to mean curvature flow self-shrinkers. result Establishes upper bounds for Betti numbers, vanishing theorem for cohomology, and dichotomy for ends.
New proof of index theorem for topological manifold bundles.
problem Index theorem for fiber bundles of compact topological manifolds.
method Use of a convenient framework for bivariant theories and recent results on the homotopy type of the topological cobordism category.
result Refinement of the assembly map for an extended A-theory characteristic.
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…
In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.
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.
New theorem links tropical phased matroids to higher-dimensional spheres.
problem Understanding topological properties of tropical phased matroids.
method Proving homeomorphism between topological order complex and a sphere.
result Topological order complex of tropical phased matroids is a (2n−3)-sphere. The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in th…
Paper extends circle pattern theory to obtuse angles.
problem Circle patterns with obtuse angles not previously covered.
method Using topological degree theory, extends Koebe-Andreev-Thurston Theorem.
result Generalized Andreev's Theorem for obtuse dihedral angles.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends topological theorems to new mapping families.
problem Limitations of existing topological theorems.
method Replaces large subset requirements with smaller subset requirements.
result Extends theorems to m-sphere to n-space mappings with m<n.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…
Stability of mapping spaces is shown to be related to the D-topology.
problem Understanding the relationship between stability and the D-topology of mapping spaces.
method Reformulation and proof of stability theorems in diffeological étale manifolds.
result Stable classes of mapping spaces are D-open.
Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.
problem Normal forms for Lagrangian submanifolds in multisymplectic geometry.
method Detailed, self-contained proof of normal form theorem, including necessary results in foliated differential topology.
result Geoffrey Martin's theorem provides a normal form for Lagrangian submanifolds in multisymplectic geometry.
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-bro…
Study on constraints for topological and smooth realizations of line arrangements and configurations.
problem Investigating constraints on topological and smooth realizations of combinatorial line arrangements and (nk)-configurations. method Exploring constraints via locally-flatly or smoothly embedded 2-spheres, using Furuta's 10/8-Theorem, and G-signature theorem.
result Established a new lower bound for (nk)-configurations, showing n≥k2−5 for topological realizations. Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.
New topological restrictions found for spaces with nonnegative Ricci curvature.
problem Understanding topological properties of spaces with nonnegative Ricci curvature.
method Analyzing complete Riemannian manifolds and RCD(0,n) spaces, applying rigidity and vanishing theorems.
result Proved a Betti number rigidity theorem and a vanishing theorem for simplicial volume.
Hanner's theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanner's original proof of the theorem is quite simple for separable spaces, it is rather involved for the general case. We provide a proof which is not only short, but a…
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.
problem Analyzing semi-characteristics on certain manifolds.
method Combines assembly maps and Hodge theorem perspectives.
result Proves an Atiyah type vanishing theorem.
New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are topologically two-spheres. Later, during the 1990s, by applying a variant of Hawking'…
The Gannon-Lee singularity theorems give well-known restrictions on the spatial topology of singularity-free (i.e., nonspacelike geodesically complete), globally hyperbolic spacetimes. In this paper, we revisit these classic results in the light of recent developments, especially the failure in higher dimensions of a c…
We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ and are topologically "glued" out of algebraic hypersurfaces in $(\C^*)^n$. Our construction can be viewed as a version of the Viro gluing theorem, relating topology of algebraic hyper…