The homology of invariant group chains
Kevin P. Knudson
如果$Q$是一个作为另一个群$G$（具有有限轨道）的自同构群的群，用$C_*(G)^Q$表示棒复形$C_*(G)$中$Q$不变链的子复形。本文研究了复体$C_*(G)^Q$的同调性，并计算了几个例子.
If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_*(G)^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_*(G)$. In this paper, we study the homology of the complex $C_*(G)^Q$ and compute several examples.
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A survey of simplicial, relative, and chain complex homology theories for hypergraphs
本文综述了从超图构造单纯、相对和链复形以建立同调理论的九种方法，并探讨其性质、函子性及小超图示例。
Invariant chains and the homology of quotient spaces
对于有限群G和有限G-CW复形X，构造了群H_ \bullet（G，X）作为胞腔链复形C_ \bullet（X）的G-不变量的同调群.这些群通过一个范数映射与商空间X/G的同调有关，从而提供了计算H_ \bullet（X/G）的机制.通过计算几个例子，给出了"Smith理论”的一个新的证明：若G=Z/p，且X是G作用于其上的模p同调球面，则子复形X^G是空的或模$p$同调球面.我们还得到了Conner猜想的一个新的证明：如果G=Z/p作用在Z-非循环空间X上，则X/G是Z-非循环的.
CoHochschild homology of chain coalgebras
推广Doi和Idrissi的工作，我们定义了任何交换环上链余代数的coHochschild同调理论，并证明了它关于链余代数的态射到强同伦的自然性。作为结果，我们得到：如果链余代数$C$的余乘自身是链余代数直到强同伦的态射，则coHochschild复形$\cohoch (C)$具有自然的余乘结构.特别地，如果$K$是一个约化单纯集，$C_{*}K$是它的正规化链复形，则$\cohoch (C_{*}K)$自然是一个同伦余结合链余代数。当$K$是单纯悬挂时，我们给出了一个简单的，明确的关于$\cohoch (C_{*}K)$的余乘公式. coHochschild复形结构是拓扑相关的。给定两个单纯映射$g,h:K\to L$，其中$K$和$L$是约化的，$C_{*}L$的coHochschild复形与$C_{*}K$中的系数的同调同构于$g$和$h$的几何实现的同伦重合空间的同调，并且这种同构考虑了余乘结构。特别是，有一个同构，尊重余乘结构，从同源的$\cohoch(C_{*}K)$到$H_{*}\op L|K|$，同源的自由循环的几何实现$K$。
Finite type invariants of rational homology 3-spheres
我们考虑由所有有理同调球面生成的有理向量空间直到保向同胚，以及在此空间上由拉格朗日保向有理同调球面置换定义的滤子.我们确定了与此过滤的分级空间的增广雅可比图的分级空间。
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain complex, or equivalently, the homology of the supremum chain complex. In this paper, we generalize the discrete Morse theory for simplicial complexes by R. Forman [5-7] and give a discrete Morse theory for hypergraphs. We use the critical simplices of the associated simplicial complex to construct a sub-chain complex of the infimum chain complex and a sub-ch…
Equivariant chain complexes, twisted homology and relative minimality of arrangements
证明了超平面排列A的补M（A）上的极小CW-结构X的等变链复形与X无关。当A是非球面排列的充分一般的线性截面时，我们给出了计算M（A）的扭曲同调的一种新方法。
Singular chain intersection homology for traditional and super-perversities
我们引入了一个奇异链交同调论，它推广了King的理论，并与Goresky和MacPherson关于任何拓扑分层伪流形的Deligne层交同调一致，无论这些拓扑分层伪流形是紧的还是不紧的，具有常系数或局部系数，并且与传统的伪流形或超伪流形（满足p（2）>0的伪流形）一致.对于p（2）=1的情形，后一种反常性是由Cappell和Shaneson引入的，并且在他们的嵌入超对偶定理中起着关键作用。我们进一步描述了这个奇异链复形的层化及其对更广泛的分层空间类的适应性。
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of Reidemeister, Yoshikawa, and Roseman moves. Our homology has a degenerate subcomplex. The normalized homology yields invariants of knots and knotted surfaces.
In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology $3$-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with $2$-torsion.
Decomposition of Persistent Homology and Spectral Sequences
研究域上过滤链复形的持续同调与谱序列关系，提出基于持续同调分解的方法，证明在一般假设下两者捕获相同信息。
We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW deco…
Morse-Bott homology
通过构造一个Morse-Bott-Smale函数的链复形，给出Morse同调定理的一个新的证明.当Morse-Smale函数为Morse-Smale函数时，该链复形化为Morse-Smale-Witten链复形;当Morse-Smale函数为常数时，该链复形化为光滑奇异$N$ -立方体链的链复形.利用不定常梯度流线的紧化模空间证明了一个Floer型延拓定理，证明了链复形的同调与Morse-Bott-Smale函数无关.
Quandle Homology Theory and Cocycle Knot Invariants
本文综述了最近发表的关于quandle同调论和上圈结不变量的几篇论文。在这里，我们描述了上循环结不变量，定义在一个状态和的形式，quandle同源性，和方法构建非平凡的上同调类。
Approximating L^2-invariants and homology growth
本文研究了贝蒂数、奇异同调的生成元的最小个数、奇异同调的挠子群的阶以及挠不变量等不变量的渐近性质.我们将证明，所有这些都消失在极限，如果CW-复杂的考虑纤维在一个特定的方式。特别地，如果考虑一个非球面闭流形，它允许一个非平凡的S^1-作用，或者它的基本群包含一个无限正规初等顺从子群，我们将证明所有这些在极限中消失。通过考虑分类空间，我们也得到了群的结果。
We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.
Szczarba's twisted shuffle and equivariant path homology of directed graphs
本文研究了标记设定下路径同调的等变性质，通过扭曲洗牌定义了链同构，并应用于有向图的Borel构造。
There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification of (invariant) group extensions.
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the …
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diag…
A cocyclic construction of $S^1$-equivariant homology and application to string topology
本文通过上循环构造建立循环同调与$S^1$-等变同调的联系，并应用于弦拓扑，给出自由环空间上重力代数结构的链级细化。
Galois descent of additive invariants
利用最近的非交换动机理论，我们证明了每个加法不变量满足伽罗瓦下降。实例包括混合复形、霍赫希尔德同调、循环同调、周期循环同调、负循环同调、连接代数K理论、模1代数K理论、非连接代数K理论、同伦K-理论、拓扑霍赫希尔德同调和拓扑循环同调。
Cofree coalgebras over operads II Homology invariance
本文证明了余生成复形的同调等价在许多有趣的情形下诱导出余自由余代数的同调等价。证明了任意余代数的基础链复形都是自由运算子上的余代数的基础链复形的直和项。结合前面的结果证明了所有余自由余代数的同调不变性。
A volume form on the Khovanov invariant
Reidemeister扭转构造可以应用于链复形，用于计算纽结或链环的Khovanov同调。这定义了一个体积形式的Khovanov同调。在Reidemeister移动下，体积形式正确地变换，以给出关于Khovanov同调的不变体积。本文论证了它的结构及其在这些移动下的不变性。此外，给出了一些例子的不变量的特殊选择的同调群的基础，以获得一个数值不变量的纽结和链接。在这些例子中，当在$\mathbb{Z}$上计算同调时，在Khovanov链复形中看到的代数扭转被恢复。
Concordance invariant $Υ$ for balanced spatial graphs using grid homology
本文将基于网格同调的组合 $Υ$ 不变量从纽结推广到平衡空间图，并借助配边给出链环间 $Υ$ 的上下界，同时证明其对纽结是配边不变量。
Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.
Homology 3-spheres in codimension three
对于积分同调3-球面在6-球面上的光滑嵌入，我们定义了一个整数不变量，它是关于它们的Seifert曲面的。我们的不变量给出了这样的嵌入和整数的光滑合痕类的集合之间的双射。它也产生了一个完整的不变量的同调边界类的所有嵌入的同调3-球面的6-球面。因此，我们表明，两个嵌入的定向积分同调3-球的6-球是同位素的当且仅当他们是同调边界。我们还将我们的不变量与Rohlin不变量联系起来，并相应地将那些可压缩的嵌入嵌入到5-球面中。
We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in earlier work of the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.
We introduce a new homology theory of uniform spaces, provisionally called $μ$-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that $μ$-homology theory satisfies the Eilenberg-Steenrod axioms. The characterization of chain-connectedness in terms of $μ$-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that $μ$-homology theory satisfies the S-homotopy axiom, and that every uniform space can be S-deformation retracted to a dense subset. It follows that for e…
For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homolog…
Finiteness of rank invariants of multidimensional persistent homology groups
秩不变量是由连续向量值函数多重过滤的空间的贝蒂数的参数化版本。本文给出了它们有限的一个充分条件。这个条件对于可嵌入R^n的空间是尖锐的。
The main target of this thesis is to solve the Perron's conjecture. This conjecture affirms that some function on the mod p Torelli group, with values in Z/p, is an invariant of mod p homology 3-spheres. In order to solve this conjecture, in this thesis we first study the mod p homology 3-spheres, the rational homology 3-spheres and those that can be realized as a Heegaard splitting with gluing map an element of the mod p Torelli group. In particular we give a criterion to determine whenever a rational homology 3-sphere has a Heegaard splitting with gluing map an element of the Torelli group m…
Three approaches to Morse-Bott homology
本文综述了用Morse-Bott函数计算有限维紧致光滑闭流形同调的三种方法，并讨论了这三种方法之间的关系。第一种方法是将函数扰动为Morse函数，第二种方法是使用级联的模空间，第三种方法是使用Morse-Bott多重复形。对于一个显式扰动（它可以用来推导Morse-Bott不等式），前两种方法得到相同的链复形直到符号。第三种方法是根本不同的。它将奇异立方链和Morse链合并在同一个复形中，提供了一种奇异立方链复形和Morse-Smale-Witten链复形之间的插值方法.
We introduce an invariant for trivalent fatgraph spines of a once bordered surface, which takes values in the first homology of the surface. This invariant is the secondary object coming from two 1-cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev, and the author in 2013. We present an explicit formula for this invariant and investigate its properties. We also show that the mod 2 reduction of the invariant is the difference of naturally defined two spin structures on the surface.
Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields
本文提出一种基于组合多向量场的持续框架，用于马尔可夫链中的Morse分解，通过转移概率定义过滤，构建多向量场并应用Morse分解，得到不变集的层次结构及持续图，以分析有限状态随机过程的复发性和稳定性。
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
G-invariant Persistent Homology
经典持久同调是形状比较的有力数学工具。不幸的是，它并不适合于研究不同于拓扑空间X的所有自同胚的群Homeo（X）的变换群的作用。这一事实限制了它在应用中的使用。为了得到与Homeo（X）的子群G相关联的自然伪距离d_G的更好的下界，我们需要调整持久同调并考虑G-不变持久同调.粗略地说，其主要思想在于通过一组在G的作用下不变的链来定义持久同调。本文将这一思想形式化，并证明了G-不变持久同调中持久贝蒂数函数关于自然伪距离d_G的稳定性。我们还展示了如何G-不变的持久同调可用于有关形状比较的应用，当不变群是一个拓扑空间的所有自同胚群的一个真子群。在本文中，我们将假设空间X是可三角化的，以保证持久贝蒂数函数是有限的，而不使用任何驯服性假设。
We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the co…
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