Tootfinder

On the Donaldson-Scaduto conjecture
Saman Habibi Esfahani, Yang Li
https://arXiv.org/abs/2401.15432 https://arXiv.org/pdf/2401.15432<…

On the Donaldson-Scaduto conjecture
Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in the $G_2$-manifold $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equa…

This https://arxiv.org/abs/2303.15981 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Čech cohomology of Morse boundaries
We consider cusped hyperbolic $n-$manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most $n-3$ and does not vanish in dimension $n-2$. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank$-1$ simple Lie groups.

On the Donaldson-Scaduto conjecture
Saman Habibi Esfahani, Yang Li
https://arXiv.org/abs/2401.15432 https://arXiv.org/pdf/2401.15432<…

On the Donaldson-Scaduto conjecture
Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in the $G_2$-manifold $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equa…

This https://arxiv.org/abs/2211.11168 has been replaced.
link: https://scholar.google.com/scholar?q=a

Total positivity in twisted product of flag varieties
We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit parameterizations of each totally positive cell. In the special cases of double flag varieties and braid varieties, we show that the totally nonnegative parts are regular CW complexes homeomorphic to closed balls. Moreover, we prove that the link of any totally n…

This https://arxiv.org/abs/2404.05930 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

All triangulations have a common stellar subdivision
We address two longstanding open problems, one originating in PL topology, another in birational geometry. First, we prove the weighted version of Oda's \emph{strong factorization conjecture} (1978), and prove that every two birational toric varieties are related by a common iterated blowup (at rationally smooth points). Second, we prove that every two PL homeomorphic polyhedra have a common stellar subdivisions, as conjectured by Alexander in~1930.

This https://arxiv.org/abs/2305.10908 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Exotic 4-manifolds with signature zero
We produce infinitely many distinct irreducible smooth 4-manifolds homeomorphic to #(2m+1)(CP^2 # -CP^2) and #(2n+1)(S^2 x S^2), respectively, for each m>3 and n>4. These provide the smallest exotic closed simply connected 4-manifolds with signature zero known to date, and in each one of these homeomorphism classes, we get minimal symplectic 4-manifolds. Our novel exotic 4-manifolds are derived from fairly special small Lefschetz fibrations we build via positive factorizations in the mapping cl…

On the topological classification of complex plane curve singularities
Alberto Fern\'andez-Hern\'andez
https://arxiv.org/abs/2402.13941 https://

On the topological classification of complex plane curve singularities
This final degree project is devoted to study the topological classification of complex plane curves. These are subsets of $\mathbb{C}^2$ that can be described by an equation $f(x,y)=0$. Loosely speaking, curves are said to be equivalent in a topological sense whenever they are ambient homeomorphic, i.e., there exists an orientation-preserving homeomorphism of the ambient space carrying one curve to the other. The project's aim is to develop operative and clear conditions to determine whether t…

Compactification of spaces of measures and pseudocompactness
Vladimir I. Bogachev
https://arxiv.org/abs/2403.15573 https://arxiv.org/…

Compactification of spaces of measures and pseudocompactness
We prove pseudocompactness of a Tychonoff space $X$ and the space $\mathcal{P}(X)$ of Radon probability measures on it with the weak topology under the condition that the Stone-Čech compactification of the space $\mathcal{P}(X)$ is homeomorphic to the space $\mathcal{P}(βX)$ of Radon probability measures on the Stone-Čech compactification of the space~$X$.

This https://arxiv.org/abs/1909.03641 has been replaced.
link: https://scholar.google.com/scholar?q=a

Definable (co)homology, pro-torus rigidity, and (co)homological classification
We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solen…

Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian
Igor Petkov, Ruslan Salimov, Mariia Stefanchuk
https://arxiv.org/abs/2404.13012

Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian
We investigate the asymptotic behavior at infinity of regular homeomorphic solutions of the nonlinear Beltrami equation with the Jacobian on the right-hand side. The sharpness of the above bounds is illustrated by several examples.

This https://arxiv.org/abs/2311.18638 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Group actions on monoidal triangulated categories and Balmer spectra
Let $G$ be a group acting on a left or right rigid monoidal triangulated category ${\mathbf K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of ${\mathbf K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of ${\mathbf K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of ${\mathbf K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantizati…

This https://arxiv.org/abs/2208.00954 has been replaced.
link: https://scholar.google.com/scholar?q=a

Spectral Spaces in o-minimal and other NIP theories
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compa…

This https://arxiv.org/abs/2210.14886 has been replaced.
link: https://scholar.google.com/scholar?q=a

Rigidity for piece-wise smooth circle maps and certain GIETs
The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is $C^1$. Mo…

This https://arxiv.org/abs/2312.01902 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Trisections of PL 4-manifolds arising from colored triangulations
The purpose of the present paper is twofold: firstly to extend to non-orientable compact 4-manifolds the notion of gem-induced trisection, directly obtained from colored triangulations (or, equivalently, from colored graphs encoding them, called gems); secondly to prove that, both in the orientable and non-orientable case, if the boundary is homeomorphic to a connected sum of sphere bundles over $\mathbb S^1$, gem-induced trisections naturally give rise to trisections of the corresponding close…

Alexandrov spaces are CS sets
Tadashi Fujioka
https://arxiv.org/abs/2404.14587 https://arxiv.org/pdf/2404.14587

Alexandrov spaces are CS sets
We prove that the extremal stratification of an Alexandrov space by Perelman-Petrunin is a CS stratification in the sense of Siebenmann. We also show that every space of directions of an Alexandrov space without proper extremal subsets is homeomorphic to a sphere. In the appendix we give an example of a primitive extremal subset of codimension 2 that is not an Alexandrov space with respect to the induced intrinsic metric.

A decomposition theorem for the affine Springer fibers
Zongbin Chen
https://arxiv.org/abs/2404.08225 https://arxiv.org/pdf/2404.08225…

A decomposition theorem for the affine Springer fibers
According to Laumon, an affine Springer fiber is homeomorphic to the universal abelian covering of the compactified Jacobian of a spectral curve. We construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to \mathcal{B}_{n}$ of the finite abelian coverings of this compactified Jacobian, and decompose the complex $Rf_{n,*}\mathbf{Q}_{\ell}$ as direct sum of intersection complexes. Pass to the limit, we obtain a similar expression for the homology of the affine Springer fibers. A qui…

This https://arxiv.org/abs/2208.00954 has been replaced.
link: https://scholar.google.com/scholar?q=a

Spectral Spaces in o-minimal and other NIP theories
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compa…

This https://arxiv.org/abs/2312.13251 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The 2-rank of finite groups acting on hyperelliptic 3-manifolds
We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set…

High-order numerical integration on regular embedded surfaces
Gentian Zavalani, Michael Hecht
https://arxiv.org/abs/2403.09178 https://

High-order numerical integration on regular embedded surfaces
We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex transformation, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the tensor-product Clenshaw-Curtis quadrature is ap…

The boundary of bordified Outer space
Karen Vogtmann
https://arxiv.org/abs/2404.11371 https://arxiv.org/pdf/2404.11371

The boundary of bordified Outer space
We study the boundary of the "Jewel space" $\mathcal J_n$ constructed in arXiv:1709.01296. This is an equivariant deformation retract of Outer space $CV_n$ on which $Out(F_n)$ acts properly and cocompactly, and is homeomorphic to the Bestvina-Feighn bordification of $CV_n$. In the current paper we analyze the structure of the boundary of $\mathcal J_n$. We then use the desctiption of the simplicial closure $CV_n^*$ as the sphere complex of a connected sum of $n$ copies of $S^1\times S^2$ to pro…

This https://arxiv.org/abs/2206.11456 has been replaced.
link: https://scholar.google.com/scholar?q=a

Metric Optimization in Penner Coordinates
Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints…

This https://arxiv.org/abs/2306.16911 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tangential smoothings of $k$-fold connected sum of complex projective spaces
This paper determines tangential structure set of $\#_k\mathbb{C}P^{n}$, for $3\leq n \leq 7$, by analyzing their stable cohomotopy groups and $KO$-groups. As a consequence, it establishes the existence of manifolds with tangential homotopy type but not homeomorphic to $\#_k\mathbb{C}P^{n}$ for $n=4,5$.

This https://arxiv.org/abs/2311.06439 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Splitting for some classes of homeomorphic and coalescing stochastic flows
The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E.~Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield est…

This https://arxiv.org/abs/2312.13251 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The 2-rank of finite groups acting on hyperelliptic 3-manifolds
We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set…

All triangulations have a common stellar subdivision
Karim Adiprasito, Igor Pak
https://arxiv.org/abs/2404.05930 https://arxiv.org/pd…

All triangulations have a common stellar subdivision
We address two longstanding open problems, one originating in PL topology, another in birational geometry. First, we prove the weighted version of Oda's \emph{strong factorization conjecture} (1978), and prove that two toric varieties that are related by a common iterated blowup (at rationally smooth points). Second, we prove that every two PL homeomorphic polyhedra have a common stellar subdivisions, as conjectured by Alexander in 1930.

The Ginsburg--Sands theorem and computability theory
Heidi Benham, Andrew De Lapo, Damir Dzhafarov, Reed Solomon, Java Darleen Villano
https://arxiv.org/abs/2402.05990

The Ginsburg--Sands theorem and computability theory
The Ginsburg--Sands theorem from topology states that every infinite topological space has an infinite subspace homeomorphic to exactly one of the following five topologies on $ω$: indiscrete, discrete, initial segment, final segment, and cofinite. The original proof is nonconstructive, and features an interesting application of Ramsey's theorem for pairs ($\mathsf{RT}^2_2$). We analyze this principle in computability theory and reverse mathematics, using Dorais's formalization of CSC spaces. …

This https://arxiv.org/abs/2107.09407 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Parabolic Mandelbrot Set
We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal e…

On four-manifolds without 1- and 3-handles
R. Inanc Baykur
https://arxiv.org/abs/2403.14586 https://arxiv.org/pdf/2403.14586

On four-manifolds without 1- and 3-handles
We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i.e. they can be given handle decompositions without 1- and 3-handles. In particular, there are many such 4-manifolds homeomorphic but not diffeomorphic to the standard 4-manifolds # m (S^2 x S^2) and # n (CP^2 # -CP^2), respectively, which answers Problem 4.91 on Kirby's 1997 list.

On metric equivalence of the Brieskorn-Pham hypersurfaces
Alexandre Fernandes, Zbigniew Jelonek, Jos\'e Edson Sampaio
https://arxiv.org/abs/2404.06922 …

On metric equivalence of the Brieskorn-Pham hypersurfaces
We show that two bi-Lipschitz equivalent Brieskorn-Pham hypersurfaces have the same multiplicities at $0$. Moreover we show that if two algebraic $(n-1)$-dimensional cones $P, R\subset\mathbb C^n$ with isolated singularities are homeomorphic, then they have the same degree.

Log-Lipschitz and H\"older regularity imply smoothness for complex analytic sets
Jos\'e Edson Sampaio
https://arxiv.org/abs/2404.06943 https://

Log-Lipschitz and Hölder regularity imply smoothness for complex analytic sets
In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of $\mathbb{C}^2$ as an algebraic surface. For instance, we prove that a complex analytic set that is log-Lipschitz regular at 0 (i.e., a complex analytic set that has a neighbourhood of the origin which bi-log-Lipschitz homeomorphic to an Euclidean ball) must be smoo…

Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds
Nikolai Erokhovets
https://arxiv.org/abs/2403.00492 h…

Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds
We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on the real moment-angle manifold $\mathbb R\mathcal{Z}_P$ corresponding to a simple convex $n$ polytope $P$ with $m$ facets. The criterion when the orbit space $\mathbb R\mathcal{Z}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb R\mathcal{Z}_P/H$ homeomorphic to $S^n$ and series …

On the homotopy type of the space of fiberings of $S^1 \times S^2$ by simple closed curves
Yi Wang, Jingye Yang
https://arxiv.org/abs/2404.08545 https://…

On the homotopy type of the space of fiberings of $S^1 \times S^2$ by simple closed curves
For most aspherical Seifert-fibered 3-manifolds $M$, the space of Seifert fiberings $SF(M)$ is known to have contractible components. It is also known that the space of Hopf fiberings of the three-sphere is noncontractible. We provide the second example of a non-aspherical 3-manifold $M$ such that $SF(M)$ has noncontractible components. In particular, we show that certain components of $SF(S^1 \times S^2)$ are homotopy equivalent to a subspace homeomorphic to the identity-based loop space $ΩSO…

Exceptional or half-integral chirally cosmetic surgeries
Kazuhiro Ichihara, Toshio Saito
https://arxiv.org/abs/2403.06701 https://arx…

Exceptional or half-integral chirally cosmetic surgeries
A pair of Dehn surgeries on a knot is called chirally cosmetic if they yield orientation-reversingly homeomorphic 3-manifolds. In this paper, we consider exceptional or half-integral chirally cosmetic surgeries, and obtain several restrictions.

Smooth Structures on $M^n\times\mathbb{S}^k$
Samik Basu, Ramesh Kasilingam, Ankur Sarkar
https://arxiv.org/abs/2402.18914 https://arx…

Smooth Structures on $M^n\times\mathbb{S}^k$
This paper explores various differentiable structures on the product manifold $M \times \mathbb{S}^k$, where $M$ is either a 4-dimensional closed oriented manifold or a simply connected 5-dimensional closed manifold. We identify the possible stable homotopy types of $M$ and use it to calculate the concordance inertia group and the concordance structure set of $M\times\mathbb{S}^k$ for $1\leq k\leq 10.$ These calculations enable us to further classify all manifolds that are homeomorphic to $\mat…

This https://arxiv.org/abs/2211.05957 has been replaced.
link: https://scholar.google.com/scholar?q=a

Linking numbers of modular knots
The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots. The linking numbers between modular knots and the trefoil is well understood. Indeed, Etienne Ghys sho…