The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.
problem Understanding fiber sum formulas for 4-manifolds.
method Using the connection between 4-manifolds and topological modular forms from 6d (1,0) SCFTs.
result Even free theories exhibit nontrivial fiber sum formulas, sensitive to individual theories and parameters.
Summing over 3-manifolds using TQFT partition functions.
problem Summing over all 3-manifolds with fixed boundary.
method Rewriting the sum over 3-manifolds as a sum over homology groups, using TQFT partition functions and topological boundary conditions.
result Existence of a distribution of 2d TQFTs whose ensemble average equals the sum over 3-manifolds.
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
The paper improves bounds on topological complexity for certain manifolds.
problem Determining bounds on topological complexity for specific manifolds.
method Analyzing cohomology classes and their pullbacks, using Gromov norm.
result Improved bounds on topological complexity for connected sums.
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the Z16 anomaly indicator for time-reversal symmetric topological superconductors. Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…
The paper studies Gauss sums and their applications in algebra and topology.
problem Computing signatures of bilinear forms and Poincaré spaces.
method Investigates properties of Gauss sums and applies them to algebraic and topological problems.
result Reproves and generalizes results on signatures mod 8 of bilinear and Poincaré spaces.
Study open 3-manifolds as sums of closed ones, finding a classification.
problem Classifying open 3-manifolds that are sums of closed 3-manifolds.
method Introduced topological invariants, classified when finitely many summands up to diffeomorphism.
result Unified classification of open 3-manifolds and closed 3-manifolds.
We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangulations, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also provide an application to state-sum invariants of 3-dimensional topological pse…
The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.
problem Understanding the homotopy type of gyrations of sphere products and connected sums.
method Recasting Fico's Lemmata into modern homotopy theoretic setting.
result Generalization of Fico's Lemmata to sphere products and connected sums.
We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.
For a closed 4-manifold X and closed 3-manifold M we investigate the smallest integer n (perhaps infinity) such that M embeds in the connected sum of n copies of X. It is proven that any lens space (or homology lens space) embeds topologically locally flatly in a connected sum of 8 copies of the complex projective plan…
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
New cosmological spacetimes without CMC Cauchy surfaces found.
problem Finding CMC Cauchy surfaces in cosmological spacetimes.
method Generalized Bartnik's construction to connected sums of three-manifolds.
result Cosmological spacetimes without CMC Cauchy surfaces for any compact three-manifolds.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is ag…
The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.
problem Understanding the geometry of convex sums of Riemannian metrics.
method Quantitative inverse function theorem and Riemannian geometry techniques.
result Injectivity radii of convex sums have uniform lower bounds.
Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.
problem Defining and understanding Murasugi sum in 4D for knotted surfaces.
method Introduced a 4D Murasugi sum to define arborescent knotted surfaces.
result Defined and studied arborescent knotted surfaces using 4D Murasugi sum.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. Paper describes a state sum formula for a graph coloring polynomial.
problem Counting n-face colorings of ribbon graphs for various n. method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
New metric reduces Weyl's energy in manifold connected sums.
problem Minimizing Weyl's energy in connected sums of manifolds.
method Interplay of Bach-flat and locally conformally flat metrics, topology of Z. result Existence of a metric with lower Weyl energy than gM. Unified framework for stability and generalization of Push-Sum in decentralized learning over directed graphs.
problem Understanding stability and generalization of Push-Sum in decentralized learning over directed networks.
method Developed a unified uniform-stability framework for SGP algorithm, incorporating imbalance-aware consistency bounds.
result Established finite-iteration stability and optimization guarantees for convex and non-convex objectives.
We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta, we compute these invariants in many cases that were previously intractable. In particular, we are now able to calculate the Yamabe invariant…
Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& …
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for these topics. The thesis uses skein theory and focues on the connected sum of cop…
Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.
problem Constructing symplectic and K-contact manifolds with specific properties.
method Developed Gompf fiber connected sum operation for symplectic orbifolds and used it to construct the required manifolds.
result Constructed a K-contact Smale-Barden manifold with specified 2-homology and sharper estimates.
We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.
In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact struc…
This note quantifies, via a sharp inequality, an interplay between (a) the characteristic rank of a vector bundle over a topological space X, (b) the Z/2Z-Betti numbers of X, and (c) sums of the numbers of certain partitions of integers. In a particular context, (c) is transformed into a sum of the readily calculable B…
A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler c…
The paper examines spaces with indecomposable Higson coronae and their properties.
problem Characterizing spaces with indecomposable Higson coronae.
method Analyzing properties of Higson coronae of non-compact metric spaces.
result Spaces with indecomposable Higson coronae are coarsely equivalent to the space of natural numbers.
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.
Example of non-smooth isotopy using K3 surfaces.
problem Non-smooth isotopy examples in 4-manifolds.
method Dehn twist along a 3-sphere in K3 surfaces' connected sum.
result First example of self-diffeomorphisms isotopic to identity but not smoothly.
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…
The paper studies relationships between twisted torsion and connected sums of 3-manifolds.
problem Understanding the topology and geometry of 3-manifolds through twisted torsion.
method Investigates the multiplicative property of twisted Reidemeister torsion and its connection to the connected sum operation.
result Establishes a relationship between the multiplicative property of twisted torsion and connected sums of 3-manifolds.
We present a construction of closed 7-manifolds of holonomy G_2, which generalises Kovalev's twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author arXiv:1505.02734 use this to e…
Classifies Morse functions on 3-manifolds made from simple building blocks.
problem Classifying Morse functions on 3-dimensional manifolds.
method Examined Morse functions on 3-manifolds represented as connected sums of Heegaard genus one manifolds.
result Found conditions for the existence of Morse functions with specific properties.
In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities chang…
New techniques create irreducible 4-manifolds with specific properties.
problem Creating irreducible 4-manifolds with specific topological and geometric properties.
method Performing various operations on irreducible simply-connected 4-manifolds, including torus surgeries, symplectic fiber sums, rational blow-downs, and Lefschetz fibrations.
result For most (e,σ) coordinates, irreducible smooth structures can be found on 4-manifolds with order two fundamental group. The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example …
New examples show unknotting numbers aren't always additive.
problem Problems with additivity of unknotting numbers.
method Provided counterexamples of knots.
result u(K_1#K_2) < u(K_1) + u(K_2) for certain knots.