L^2-Betti Numbers of Discrete Measured Groupoids

The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action

The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (Ellis …

Dynamical correspondences of $L^2$-Betti numbers

We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the $L^2$-Betti numbers exactly measure the failure of addition formula for dynamical invariants.

Deligne Cohomology for Orbifolds, discrete torsion and B-fields

In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth étale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local in the case of an orbifold.

广义系数和梅耶-维托里斯对于 ample 群体的摩尔同调

Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids

本文建立了 ample 群体摩尔同调的广义系数定理和梅耶-维托里斯长正合列，通过链级证明提供了有效计算工具，强调系数群离散性的重要性。

Graph colouring and the total Betti number

The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme was proved by Bonamy, Charbit and Thomasse. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. The main technique is discrete Morse theory and building poset maps. Ramanujan graphs with arbitrary chromatic number and girth log(n) is a classical construction. We show that any subgraph of them with le…

Orbifold Euler characteristics of non-orbifold groupoids

For a finitely presented discrete group $Γ$, we introduce two generalizations of the orbifold Euler characteristic and $Γ$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The $Γ$-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the $Γ$-inertia Euler characteristic is the usual Euler characteristic of the $Γ$-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces …

构造具有固定Betti数的单项式理想

Building monomial ideals with fixed betti numbers

针对单项式理想生成元增多导致关系计算复杂度上升的问题，本文利用离散同伦论的初等坍塌构造无穷多具任意生成元数、Betti数相似或相同的单项式理想，并证明理想扩张时其无混分量的Cohen-Macaulay性质保持不变。

非负曲率图的 Betti 数估计

Betti number estimates for non-negatively curved graphs

本文研究非负 Ollivier 与 Bakry–Émery 曲率图的第一 Betti 数上界及等号刚性，并将结论推广至不可逆 Markov 链和局部例外情形。

群胚同调和分类空间同调不同构：康托尔单位群胚

Groupoid Homology and Classifying-Space Homology Are Not Isomorphic

本文通过康托尔集上的单位群胚，证明了Matui型群胚同调与分类空间奇异同调在0阶上不同构，前者为可数群而后者基数为2^ℵ₀。

Polynomial hierarchy, Betti numbers and a real analogue of Toda's theorem

Toda proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class $\mathbf{P}^{#\mathbf{P}}$, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class $#\mathbf{P}$. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines) has been missing so far. In thi…

The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study pr…

Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types

In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers…

论Betti谐余积与分圆滤过的相容性

On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations

本文证明离散Betti谐余积对任意N均与分圆滤过结构相容，从而为其完备化作为此前构造的完备Hopf代数余积的显式实现提供了候选。

用于拓扑约束优化的霍奇谱替代方法

Hodge Spectral Surrogates for Topology-Constrained Optimization

提出基于霍奇谱松弛的微分框架，通过低通谱滤波器实现拓扑约束优化，克服贝蒂数和持续同调的离散组合困难。