Mokhtar Aouina
This paper studies the homotopy type of the moduli space of compact n-manifold thickenings of a finite complex. The main result computes the homotopy fibers of the stabilization map from n-thickenings to (n+1)-thickenings in a wide range. In particular, we obtain an EHP-type sequence for thickenings extending work of C.T.C. Wall.
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We define a space of relative embedded thickenings of a given map from a finite complex to a Poincare Duality space, and show that there is a highly connected stabilization map between such spaces induced by fiberwise suspension. As a result, we obtain a generalization to the Poincare Duality category of a smooth stabilization theorem of Connolly and Williams.
Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and Čech complexes of $X/G$. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris-Rips and Čech thickenings of projective spaces a…
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the ci…
Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds …
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris complex contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, equipped with an optimal transport metric. We show that the Vietoris metric thickening and the Vietoris complex have isomorphic homotopy groups in all dimensions.…
Reeb Graph of Sample Thickenings
本文提出了一种基于Reeb图的样本膨胀重建框架，通过推广交织距离理论，证明在特定条件下样本膨胀的Reeb图能近似底层空间的Reeb图，并给出了计算方法。
Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thick…
The Persistent Topology of Optimal Transport Based Metric Thickenings
本文引入了配备最优传输度量的 $p$-Vietoris-Rips 和 $p$-Čech 度量加厚，证明了其持续同调的稳定性定理，并在 $p=2$ 时推导出 $n$-球面加厚的完整同伦类型。
Vietoris thickenings and complexes are weakly homotopy equivalent
本文解决了 Vietoris--Rips 复形与其度量加厚之间同伦类型的关系问题，证明了对于任意度量空间 $X$ 和阈值 $r>0$，两者间的自然双射构成弱同伦等价。
Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when $Y$ is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the \emph{Vietoris--Rips} and \emph{Čech thickenings}, which are equipped with the 1-Wasserstein metric in place of the simplicia…
Bond thickenings of the simplicial boundary of Outer space
该研究通过图论连通性定义的加厚，证明Culler-Vogtmann外空间的单纯边界∂ℱ𝒮到含非3边连通稳定图的子复形C'的包含映射是(2n-3)-连通的，为相关拓扑猜想提供了证据并统一了部分图复形结果。
Shifted Lagrangian thickenings of shifted Poisson derived schemes
本文证明了移位泊松结构在导出方案上的空间等价于移位拉格朗日增厚空间，解决了移位泊松几何中的猜想，并扩展了已知的移位辛结构定理至移位泊松结构。
Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian mani…
Relative Smooth Surgery Structure Sets of Thickenings of the Cayley Projective Plane and Applications
研究凯莱射影平面加厚\(\mathbb{OP}^{2}\times\mathbb{D}^{k}\)（\(k\geq1\)且\(k\equiv0\pmod4\)）的相对光滑手术结构集，通过确定法不变量等计算，有构造不同胚流形、计算同伦群、构造特殊丛等应用。
Anti-Vietoris--Rips metric thickenings and Borsuk graphs
该研究定义了反Vietoris-Rips度量加厚空间$\rm{AVR^m}(X;r)$，分析了球面的加厚空间拓扑性质（同胚、同伦等价、可缩性），给出紧黎曼流形加厚空间的覆盖维数与上同调消失条件，并建立了与Borsuk图色数的联系。
We consider moduli spaces of $d$-dimensional manifolds with embedded particles and discs. In this moduli space, the location of the particles and discs is constrained by the $d$-dimensional manifold. We will compare this moduli space with the moduli space of $d$-dimensional manifolds in which the location of such decorations is no longer constrained, i.e. the decorations are decoupled. We generalise work by Bödigheimer--Tillmann for oriented surfaces and obtain new results for surfaces with different tangential structures as well as to higher dimensional manifolds. We also provide a generalisa…
The stable cohomology of the moduli space of curves with level structures
本文证明在稳定范围内，带水平结构的曲线模空间与普通曲线模空间的有理上同调相同。
We study the moduli and determine a homotopy type of the space of all generalized Morse functions on d-manifolds for given d. This moduli space is closely connected to the moduli space of all Morse functions studied in the paper math.AT/0212321, and the classifying space of the corresponding cobordism category.
We study the moduli space of A-infinity structures on a topological space as well as the moduli space of A-infinity-ring structures on a fixed module spectrum. In each case we show that the moduli space sits in a homotopy fiber sequence in which the other terms are representing spaces for Hochschild cohomology.
In this paper we study topology of moduli spaces of tropical curves of genus $g$ with $n$ marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\it moduli space of metric graphs with $n$ marked points.} We describe the shrinking bridges strong deformation retraction, which leads to a substantial simplification of all these moduli spaces. In the rest of the paper, that reduction is used to analyze the case of genus 1. The corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation ${\mathbb Z}_2$-action; and f…
We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring.
In this paper we show that the moduli space of framed torsion-free sheaves on a certain class of toric surfaces admits a filtrable Bialynicki-Birula decomposition determined by the torus action. The irreducibility of this moduli space follows immediately as a corollary. Moreover, using its filtrable decomposition we show that the moduli space shares the same homotopy type of an invariant proper compact subvariety having the same fixed points set. We start our study from the moduli space of framed torsion-free sheaves on the projective plane. Afterwards, we generalize the results to toric surfa…
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $\leq 6$. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.
Homology stability for asymptotic monopole moduli spaces
该研究证明了带框狄拉克磁单极和理想磁单极两类渐近模空间的同调稳定性，通过配置空间的一般同调稳定性结果实现，并揭示了它们作为经典模空间部分紧化边界超曲面的结构。
We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of curves that are defined in terms of Weierstrass points.
We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both models.
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