Paul D. Mitchener
In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of `coarse CW-complexes'. This uniqueness result is used to prove a version of the coarse Baum-Connes conjecture for such spaces.
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This article corrects two mistakes in the article "Coarse homology theories" [math.AT/0106183].
We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.
We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic $K$-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to p…
Localization for coarse homology theories
本文引入Bredon风格等变粗同调理论，证明其满足局部化定理，并指出一般等变粗同调理论可被其逼近，进而得到Segal局部化定理的粗类比。
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly map explicitly in terms of locally finite homology theory.
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\mathbf{BornCoarse}$ to the cocomplete stable $\infty$-category of chain complexes $\mathbf{Ch}_\infty$. We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\mathcal{X} H^G$ by constructing a trace-like natural transformation $\mathcal{X} K_{k}^G\to…
In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M. C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord's and our homology are isomorphic for all compact uniform spaces and that Korppi's and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi's homology. As an example, we outline a proof of the continuity…
I discuss the problem of realizing families of complex orientable homology theories as families of commutative ring spectra, including a recent result of Jacob Lurie emphasizing the role of p-divisible groups.
Breaking symmetries for equivariant coarse homology theories
本文提出粗几何中的对称性破缺构造，通过限制到更小群和空间获取等变粗同调类信息，并在等变粗K-同调中给出解析解释，应用于不变微分算子谱理论。
Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors to generalized homology and cohomology theories. In this paper, we construct a bivariant functor such that the associated cohomology is the Cech cohomology and the homology is the Steenrod homology (at least for compact metric spaces).
A survey of simplicial, relative, and chain complex homology theories for hypergraphs
本文综述了从超图构造单纯、相对和链复形以建立同调理论的九种方法，并探讨其性质、函子性及小超图示例。
Coarse homology theories and finite decomposition complexity
本文通过粗同调理论探讨了具有等价有限分解复杂性的空间中forget-control映射纤维的消失性质。
Pirashvili--Richter-type theorems for the reflexive and dihedral homology theories
本文证明自反同调与二面体同调可视为非交换集合范畴上的函子同调，从而建立了对应理论的 Pirashvili–Richter 型定理。
Semi-coarse Spaces, Homotopy and Homology
本文研究半粗空间的代数拓扑，构造同伦等价下不变的同调群，并证明无向图诱导的半粗同调与 Vietoris-Rips 同调同构，导出同伦不变性定理。
Topological equivariant coarse K-homology
本文构造了具有严格群作用的$C^{*}$-范畴上的等变粗同调理论，通过引入受控于有界粗空间的Roe范畴并应用同调函子，验证了轨道范畴上的CP-函子性质，进而影响装配映射的单射性。
The tools and arguments developed by Kevin Costello are adapted to families of "Outer Spaces" or spaces of graphs. This allows us to prove a version of Deligne's conjecture: the Harrison homology associated to a homotopy commutative algebra is naturally a module over a particular cobordism category of 3-manifolds.
We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all,…
We introduce the notion of the space of parallel strings with partially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to construct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids.
We derive the Pimsner-Voiculescu sequence calculating the K-theory of a $C^{*}$- algebra with $\mathbb{Z}$-action using constructions with equivariant coarse K-homology theory. We then investigate to which extend this idea extends to more general equivariant coarse homology theories.
Injectivity results for coarse homology theories
本文通过等效粗同调理论和降维原理，研究了线性群及具有有限分解复杂度群的装配映射单射性问题。
We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped with various tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Factorization homology theories for singular manifolds are characterized by a generalization of the Eilenberg-Steenrod axioms. Using these axioms, we exte…
We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S -> A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.
Transgressions and Chern characters in coarse homotopy theory
本文探讨了粗同调论中超越现象与 Chern 特征类之间的关系，研究了它们之间的可交换性及数学结构。
In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these constructions is that there are corresponding Poincaré dual (co)homology theories called backwards (co)homology. This gives rise to a third relative (co)homology theory which we call stratifold Tate (co)homology. These Tate groups agree with the original definition of Tate cohomology for finite groups given by Swan. All constructions in this paper are geometric…
Primitive path homology
本文在简单有向图范畴引入原始路径同调理论，其在非对称有向图子范畴与已有路径同调理论一致但一般情况不同，还可构造带指定头尾顶点的路径原始同调并研究其与路径同调的关系。
Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required.On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable for…
In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca & Ege. We show that the two simplicial homology theories are isomorphic, and that the two cubical theories are isomorphic, but that the simplicial theory is not isomor…
A cubical approach to homology theories for hypergraphs
本文提出超图的三种同调理论（Γ、□、×），证明其互异且不同于嵌入同调，并引入扩展图离散同伦的超图同伦，发现仅□-同调具同伦不变性，且Γ-同调有独特结构行为。
In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from $f_s$ and $h$, and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will …
We show that, if E is a Landweber exact ring spectrum, then the category of E_*E-comodules is equivalent to the localization of the category of BP_*BP-comodules with respect to the hereditary torsion theory of v_n-torsion comodules, where n is the height of E. In particular, the category of E(n)_*E(n)-comodules is equivalent to the category of (v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems for E_*E-comodules; we show every E_*E-comodule has a primitive, we classify the invariant radical ideals, and we prove a version of the Landweber filtration theorem.
We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $\mathrm{K}$-theory and -homology, the secondary products …
We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to…
Coarse cohomology of the complement
该研究定义了度量空间子空间补集的粗（上）同调以推广Roe的理论，构建补集粗几何结构的模型空间，提出粗庞加莱对偶空间新研究方法，证明粗亚历山大对偶版本并给出粗PD($n$)空间的同调判别准则。
Branched coarse coverings and transfer maps
本文引入分支粗覆盖和转移映射概念，证明多种粗K-同调理论具有转移结构，并利用粗同伦理论中的Atiyah L²指标定理为Higson的粗Baum-Connes猜想反例提供新论证。
The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there ar…
To a homology theory one can associate an additive site and a new homological functor with values in the category of additive sheaves on that site. If this category of sheaves can be shown to be equivalent to a category of comodules of a Hopf algebroid, then we obtain a new homology theory by composing with the underlying module functor. This new homology theory is always flat and we call it a flat replacement of the original theory. For example, Pstrągowski has shown that complex cobordism is a flat replacement of singular homology. In this article we study the basic properties of the sites a…
On MU-homology of connective models of higher Real K-theories
本研究利用切片过滤分析高阶实K理论连通模型不动点的MU同调，揭示其与周期版本的差异，明确偶无挠性成立的高度条件，并完成tmf₀(3)对应余模代数的完整计算。
Torsion in Magnitude homology theories
研究有限图magnitude同调与Eulerian magnitude同调的结构关系，给出Eulerian magnitude同调存在挠元的两种证明、挠元类型与阶的分析及多类图的显式计算。
On the Universal Coefficient Formula and Derived $\varprojlim ^{(i)} $ Functor
本文研究同调与逆向极限不交换问题，在一般拓扑空间上定义正合同调函子，给出其与其他经典理论的关系，并证明其紧致性与连续性。
Magnitude meets persistence. Homology theories for filtered simplicial sets
本文探讨了幅度同调与持续同调在计算过滤简形集同调中的不同方法，通过研究有限度量空间的尺度变化，揭示两者在拓扑数据分析中的关联。
In this article we develop the cotangent complex and (co)homology theories for spectral categories. Along the way, we reproduce standard model structures on spectral categories. As applications, we show that the invariants to descend to stable $\infty$-categories and we prove a stabilization result for spectral categories.
In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of …
Coarse geometry
本文综述粗几何，着重介绍粗同调理论这一核心方法，为该领域提供系统性概述。
This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg--Steenrod axioms; it can also be viewed as an analogue of the Baez--Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topologic…
In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type.
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…
A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop technique for the description of the higher limits over categories of presentations and show that certain homological functors can be described in this way. In particular, we give a description of Hochschild homology and the derived functors of tensor, symmetric and exterior powers in the sense of Dold and Puppe as higher limits.
In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that …
In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.
Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\Or(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair $(G,H)$ defined by Adamson and Takasu to have coefficients in an $\Or(G)$-module. There is a canonical comparison homomorphism defined by Cisneros-Molina and Arciniega-Nevárez from Takasu's theory to Adamson's one. We give a necessary and sufficient condition on the subgroup $H$ for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a …
Homotopy, homology, and persistent homology using closure spaces
本文在 Čech 闭包空间滤过上构建多种同伦与奇异同调理论以生成持续模，并推广 Gromov-Hausdorff 距离证明其稳定性。
These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.
We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group $G$. As coefficients, we allow $E_\infty$-algebras with $G$-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the $(\infty,1)$-category of cospans of $E_\infty$-algebras.
For commutative algebras there are three important homology theories, Harrison homology, Andre-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quille…
Persistent Homology of Coarse Grained State Space Networks
本文提出利用粗粒化状态空间网络结合持久同调进行动态状态检测，相比序数划分网络和时间延迟嵌入方法，在检测性能、噪声鲁棒性和计算效率上均有显著优势。
We propose the notion of a coarse cohomology theory and study the examples of coarse ordinary cohomology, coarse stable cohomotopy and of coarse cohomology theories obtained by dualizing coarse homology theories. We show that the dualizing spectrum of a finitely generated torsion-free group only depends on the coarse motivic spectrum represented by the underlying bornological coarse space of the group. This in particular implies a conjecture of J. R. Klein that the dualizing spectrum of a group is a coarse invariant.
This text is based on lectures given by authors in summer 2015. It contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of non-additive functors in a form of derived limits. The theory of so-called ${\bf fr}$-codes also is developed. This is a method how different functors from the category of groups to the category of abelian groups, such as group homology, tensor products of abelianization, can be coded as sentences in the alphabet with two symbols ${\bf f}$ and ${\bf r}$.
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is known that the category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over a field k is an abelian category. Denote the category by H. In this paper, we give some ways to construct H-valued homology theories. As a main result, we give H-valued homology theories whose coefficients are neither group Hopf algebras nor function Hopf…
In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Caprano, and White, is based on graph maps from hypercubes to the graph. The second theory was developed by Grigor'yan, Lin, Muranov, and Yau, and is based on paths in the graph. Results in both settings imply that the respective homology groups are isomorphic in homological dimension one. We show that, for several infinite classes of graphs, the two theories lead to isomorphic homology groups in all dimensions. However, we provide an example for which the hom…
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