Fundamental Group Algorithm for low dimensional tessellated CW complexes

单纯复形的基群

Fundamental groups of small simplicial complexes

本文提供了至多 $8$ 个顶点的单纯复形基群的完整列表，并给出了 $9$ 个顶点的部分示例，推动了流形三角剖分猜想与不变量计算等应用。

Fixed Point Sets and the Fundamental Group I: Semi-free Actions on G-CW-Complexes

Smith theory says that the fixed point of a semi-free action of a group $G$ on a contractible space is ${\bb Z}_p$-acyclic for any prime factor $p$ of $G$. Jones proved the converse of Smith theory for the case $G$ is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain $K$-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the $K$-theoretical obstruction.

Geometry and Topology of Random 2-complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the strong hyperbolicity property of random 2-complexes.

Large Random Simplicial Complexes, II; the fundamental groups

In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domai…

Generalization of neighborhood complexes

We introduce the notion of r-neighborhood complex for a positive integer r, which is a natural generalization of Lovasz neighborhood complex. The topologies of these complexes give some obstructions of the existence of graph maps. We applied these complexes to prove the nonexistence of graph maps about Kneser graphs. We prove that the fundamental groups of r-neighborhood complexes are closely related to the (2r)-fundamental groups defined in the author's previous paper.

Poincare Duality Complexes with Highly Connected Universal Covers

Baues and Bleille showed that, up to oriented homotopy equivalence, a Poincare duality complex of dimension $n \ge 3$ with $(n-2)$-connected universal cover, is classified by its fundamental group, orientation class and the image of its fundamental class in the homology of the fundamental group. We generalise Turaev's results for the case $n=3$, by providing necessary and sufficient conditions for a triple $(G, ω, μ)$, comprising a group, $G$, $ω\in H^{1}(G;\mathbb{Z}/2\mathbb{Z})$ and $μ\in H_{3}(G, \mathbb{Z}^ω)$ to be realised by a Poincaré duality complex of dimension $n$ with $(n-2)$-conn…

Higher chordality: From graphs to complexes

We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.

$r$-fundamental groups of graphs

In this paper, we introduce the notions of $r$-fundamental groups of graphs, $r$-covering maps, and $r$-neighborhood complexes of graphs for a positive integer $r$. There is a natural correspondence between $r$-covering maps and $r$-fundamental groups as is the case of the covering space theory in topology. We can derive obstructions of the existences of graph maps from $r$-fundamental groups. Especially, $r$-fundamental groups gives deep informations about the existences of graph maps to odd cycles. For example, we prove the Kneser graph $K_{2k+1,k}$ has no graph maps to $C_5$. $r$-neighborho…

Twisted Blanchfield pairings and symmetric chain complexes

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted Blanchfield pairing is then associated to a 3-manifold together with a decomposition of its boundary into two pieces and a unitary representation of its fundamental group.

Digital Fundamental Groups and Edge Groups of Clique Complexes

In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex and its edge group is well-known to be isomorphic to the ordinary (topological) fundamental group of its geometric realization. This identification of our intrinsic digital fundamental group with a topological fundamental group---extrinsic to the digital setting---means that many familiar facts abou…

Vertex numbers of simplicial complexes with free abelian fundamental group

We show that the minimum number of vertices of a simplicial complex with fundamental group $\mathbb{Z}^{n}$ is at most $O(n)$ and at least $Ω(n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation $\langle S | R\rangle \cong \mathbb{Z}^{n}$ whose relations are of the form $g^{a}h^{b}i^{c}$ for $g, h, i \in S$ has at least $Ω(n^{3/2})$ generators.

Steenrod coalgebras III. The fundamental group

In this note, we extend earlier work by showing that if $X$ and $Y$ are delta-complexes (i.e. simplicial sets without degeneracy operators), a morphism $g:N(X)\to N(Y)$ of Steenrod coalgebras (normalized chain-complexes equipped with extra structure) induces one of 2-skeleta $\hat{g}:X_{2}\to Y_{2}$, inducing a homomorphism $π_{1}(\hat{g}):π_{1}(X)\toπ_{1}(Y)$ that is an isomorphism if $g$ is an isomorphism. This implies a corresponding conclusion for a morphism $g:C(X)\to C(Y)$ of Steenrod coalgebras on unnormalized chain-complexes of simplicial sets.

Finite group actions on Kan complexes

We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the cohomology of the fixed point set of a finite $p-$group action on an one vertex aspherical Kan complex.

Fundamental groups of neighborhood complexes

The neighborhood complexes of graphs were introduced by Lovász in his proof of the Kneser conjecture. He showed that a certain topological property of $N(G)$ gives a lower bound for the chromatic number of $G$. In this paper, we study a combinatorial description of the fundamental groups of the neighborhood complexes. For a positive integer $r$, we introduce the $r$-fundamental group $π_1^r(G,v)$ of a based graph $(G,v)$ and the $r$-neighborhood complex $N_r(G)$ of $G$. The $1$-neighborhood complex is the neighborhood complex. We show that the even part $π_1^{2r}(G,v)_{ev}$, which is a subgrou…

Automaton groups and complete square complexes

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually…

Normalisation for the fundamental crossed complex of a simplicial set

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the {\it fundamental crossed complex} of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with so…

Algebraic and combinatorial codimension-1 transversality

The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.

Indecomposable $PD_3$-complexes

We show that if $X$ is an indecomposable $PD_3$-complex and $π_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge groups are $Z/2Z$ and all but at most one of the vertex groups is dihedral of order $2m$ with $m$ odd. Every such group is realized by some $PD_3$-complex. We also propose a strategy for tackling the question of whether every $PD_3$-complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold.

Inverse monoids and immersions of 2-complexes

It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathc…

Connectedness at infinity of systolic complexes and groups

By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by euclidean space of dimension at least three. We also study semistability at infinity for some systolic groups.

Topology of random 2-complexes

We study the Linial--Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for $p\ll n^{-1}$ a random 2-complex $Y$ collapses simplicially to a graph and, in particular, the fundamental group $π_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that, if the probability parameter $p$ satisfies $p\gg n^{-1/2+ε}$, where $ε>0$, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as $n\to \infty$. We also establish several related results, for example we show…

图的闭邻域复形

Closed neighborhood complexes of graphs

研究图的闭邻域复形与其在拓扑学中的性质，揭示其与独立复形及路径同调的关系。

随机旗复形同调的单侧锐阈值

One-sided sharp thresholds for homology of random flag complexes

本文研究随机旗复形同调与基本群的概率阈值，证明存在同调不消失概率介于零与一之间的中间区间，并给出基本群为自由群的锐阈值新界，同时构造出基本群既非自由群也不具有 Kazhdan 性质 (T) 的概率区间。

Wagoner's Complexes Revisited

We generalize Wagoner's representation of the automorphism group of a two-sided subshifts of finite type as the fundamental group of a certain CW-complex to groupoids having a certain refinement structure. This significantly streamlines the original proof and allows us to extend this construction to, e.g., the automorphism group of subshifts of finite type over arbitrary finitely generated groups and the automorphism group of G-SFTs.

A connection between cellularization for groups and spaces via two-complexes

Let $M$ denote a two-dimensional Moore space (so $H_2(M; \Z) = 0$), with fundamental group $G$. The $M$-cellular spaces are those one can build from $M$ by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The question we address here is to characterize the class of $M$-cellular spaces by means of algebraic properties derived from the group $G$. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.

On homotopy types of Euclidean Rips complexes

The Rips complex at scale r of a set of points X in a metric space is the abstract simplicial complex whose faces are determined by finite subsets of X of diameter less than r. We prove that for X in the Euclidean 3-space R^3 the natural projection map from the Rips complex of X to its shadow in R^3 induces a surjection on fundamental groups. This partially answers a question of Chambers, de Silva, Erickson and Ghrist who studied this projection for subsets of R^2. We further show that Rips complexes of finite subsets of R^n are universal, in that they model all homotopy types of simplicial co…

Weighted Fundamental Group

In this paper, we develop and study the theory of weighted fundamental groups of weighted simplicial complexes. When all weights are 1, the weighted fundamental group reduces to the usual fundamental group as a special case. We also study weighted versions of classical theorems like van Kampen's theorem. In addition, we also investigate the abelianization, lower central series and applications of weighted fundamental groups.

Toric complexes and Artin kernels

A simplicial complex L on n vertices determines a subcomplex T_L of the n-torus, with fundamental group the right-angled Artin group G_L. Given an epimorphism χ\colon G_L\to \Z, let T_L^χbe the corresponding cover, with fundamental group the Artin kernel N_χ. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^χwith coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^χ;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^χ;\k…

Cohomological and geometric invariants of simple complexes of groups

We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and gi…

The fundamental group of the $p$-subgroup complex

We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $π_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in the literature of a $p$-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, $π_1(A_p(G)) = π_1(A_p(S_G)) * F$, where $F$ is a free group and $π_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = Ω_1(G)/O_{p'}(Ω_1(G))$. This result essentially reduces the study of the fundamental group of …

解构奥斯特兰德公式，I. 与加法函子相关的基本序列

Deconstructing Auslander's formulas, I. Fundamental sequences associated with additive functors

该研究针对加法函子构造长正合序列，半正合函子所得序列正合，一般函子非平凡同调可能仅在导出函子处。专门针对有限表现模上的Hom和张量积函子恢复奥斯特兰德公式，结果适用于任意环和模，还得出任意复形同调与上同调的通用系数定理。

稳定自由模与2-复形的不稳定分类

Stably free modules and the unstable classification of 2-complexes

本文证明了对于所有$k \ge 2$存在秩为$k$的非自由稳定自由$\mathbb{Z} G$-模，并以此构造出同调不同且欧拉示性数超出极小值$k$的有限2-复形，解决了Wall问题D5，并探讨了在闭光滑4-流形拓扑中的潜在应用。

Topology of random 2-dimensional cubical complexes

We study a natural model of random 2-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result is to exhibit a sharp threshold p=1/2 for homology vanishing as $n \to \infty$. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing the connectivity threshold for random cubical graphs. Our main result can also be seen as a cubical counterpart to the Linial--Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhi…

复形上的群作用、Kozsul模型、展示与Coxeter的一个定理

Group actions on complexes, Kozsul models, presentations, and a theorem of Coxeter

本文基于Kozsul思想重新证明了Brown关于离散群在单连通复形上作用可构造模顶点稳定子展示的定理，并通过基本群、对称群及二十面体相关群等例子展示其简化过程。

Poincare Duality Pairs in Dimension Three

We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for Poincare duality complexes in dimension three to the relative case of Poincare duality pairs. The results for Poincare duality complexes are recovered by restricting the results to the case of Poincare duality pairs with empty boundary. Up to oriented homotopy equivalence, three-dimensional Poincare duality pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the d…

$\mathfrak{S}$-coalgebras determine fundamental groups

In this paper, we extend earlier work by showing that if $X$ and $Y$ are simplicial complexes (i.e. simplicial sets whose nondegenerate simplices are determined by their vertices), an isomorphism $\mathcal{C}(X)\cong\mathcal{C}(Y)$ of coalgebras over the Barratt-Eccles operad implies that the 3-skeleton of $X$ is weakly equivalent to the 3-skeleton of $Y$, also implying that $π_{1}(X)=π_{1}(Y)$.

可缩2-复形上的群作用 II

Group actions on contractible $2$-complexes II

本文第二部分证明特定PSL₂(q)群作用下无环2-复形基本群有非平凡酉表示，从而完成有限群在有限可缩2-复形上的作用必存在不动点这一结果的证明。

Sharp vanishing thresholds for cohomology of random flag complexes

For every $k \ge 1$, the $k$th cohomology group $H^k(X, \Q)$ of the random flag complex $X \sim X(n,p)$ passes through two phase transitions: one where it appears, and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group $π_1(X)$ to have Kazhdan's property (T). Combining with earlier results, we obtain as a corollary that for every $k \ge 3$ there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of $k$-dimensional spheres.

Principal bundles on 2-dimensional CW-complexes with disconnected structure group

Given any topological group $G$, the topological classification of principal $G$-bundles over a finite CW-complex $X$ is long-known to be given by the set of free homotopy classes of maps from $X$ to the corresponding classifying space $BG$. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when $X$ has dimension $2$, it seems there is a case in which such explicit classification has not been explicitly considered. This is the case where $G$ is a Lie group, whose group of components acts non-trivially on its funda…

Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes

A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $\tilde{H}_{d-1}(T;\mathbb{F})=0$. The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycle…

The asphericity of random 2-dimensional complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex $Z$ obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex $Z$ satisfies the Whitehead conjecture, i.e. any subcomplex $Z'\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee...\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We a…

可缩2-复形上的群作用 I

Group actions on contractible $2$-complexes I

该系列首篇文章发展相关理论，通过构造非平凡表示证明有限群G在有限可缩2-复形上的作用有不动点，并覆盖G=PSL₂(2ⁿ)、PSL₂(q)（q≡3 mod8）或Sz(2ⁿ)的情形。

射影模与 $(G,n)$-复形的同伦分类

Projective modules and the homotopy classification of $(G,n)$-complexes

本文证明了在群具有周期上同调时，有限 $(G,n)$-复形的同伦型与特定射影 $\mathbb{Z} G$-模稳定类在自同构群作用下的轨道一一对应，并开发了显式计算该作用的技术。

Torus actions, combinatorial topology and homological algebra

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The …

刚性化单纯复形与实现群作用

Rigidifying simplicial complexes and realizing group actions

本文证明有限群在有限展示群及有限生成阿贝尔群上的作用均可实现为某空间自同伦等价群在同伦群上的作用，并证明任意有限连通单纯复形可在保持同伦型的前提下刚性化，使其自同构群约化为任意给定子群。