Homotopy and homology of fibred spaces

A-homology, A-homotopy and spectral sequences

Given a CW-complex A we define an `A-shaped' homology theory which behaves nicely towards A-homotopy groups allowing the generalization of many classical results. We also develop a relative version of the Federer spectral sequence for computing A-homotopy groups. As an application we derive a generalization of the Hopf-Whitney theorem.

Effective homology for homotopy colimit and cofibrant replacement

We extend the notion of simplicial set with effective homology to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I} \to \mathsf{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mathsf{hocolim} X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\mathsf{cof}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

Persistent Homology and Applied Homotopy Theory

This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence, vectorization of persistence barcodes, as well as some applications.

On the homotopy classification of spaces by the fixed loop space homology

Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty $) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1, $ a complete homotopy invariant is constructed in terms of the loop space homology $H_*(ΩX).$ This allows us to classify all such $R$-local spaces up to homotopy with a fixed loop space homology.

Interaction homotopy与interaction homology

Interaction homotopy and interaction homology

本文提出一个代数拓扑框架研究空间之间的相互作用，引入interaction spaces并探讨其同伦、奇异同调与单纯同调，证明interaction奇异同调是interaction同伦下的不变量。

相交同调与同伦群的关系

Relation between intersection homology and homotopy groups

本文定义了基于Gajer单纯集的相交同伦群，证明了其基本性质、CS集上的Hurewicz定理与拓扑不变性，并指出最高奇异性下其与空间自身同伦群一致。

Homotopy groups of certain highly connected manifolds via loop space homology

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M)$. Moreover, we show that these $p$-local homotopy groups can be expressed as direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work.

Simplicial Homotopy Theory, Link Homology and Khovanov Homology

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…

The stable homotopy classification of $(n-1)$-connected $(n+4)$-dimensional polyhedra with 2 torsion free homology

In this paper, we study the stable homotopy types of $\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.

A Functor Converting Equivariant Homology to Homotopy

In this paper, we prove an equivariant version of the classical Dold-Thom theorem. Associated to a finite group, a CW-complex on which this group acts and a covariant coefficient system in the sense of Bredon, we functorially construct a topological abelian group by the coend construction. Then we prove that the homotopy groups of this topological abelian group are naturally isomorphic to the Bredon equivariant homology of the CW-complex. At the end we present several examples of this result.

Rational homology and homotopy of high dimensional string links

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high dimensional anologues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher order Hochschild homology also called Hochschild-Pirashvili homology. In this paper, we generalize all these results to high dimensional analogues of spaces of string links. The methods of our paper are applicable in the ran…

度量纤维化的幅度同调与同伦型

Magnitude homology and homotopy type of metric fibrations

本文证明具有共同基空间和共同纤维的度量纤维化具有同构的幅度同调及相同的幅度同伦型，推广了 Leinster 的幅度乘积结果，并建立了幅度同伦型与 Hepworth-Willerton 带点单纯集几何实现的等价性。

Homotopy completion and topological Quillen homology of structured ring spectra

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras an…

An Interpolation between Homology and Stable Homotopy

By considering labeled configurations of ``bounded multiplicity'', one can construct a functor that fits between homology and stable homotopy. Based on previous work, we are able to give an equivalent description of this labeled construction in terms of loop space functors and symmetric products. This yields a direct generalization of the May-Milgram model for iterated loop spaces, and answers questions of Carlsson and Milgram posed in the handbook. We give a classifying space formulation of our results hence extending an older result of Segal. We finally relate our labeled construction to a t…

自由环路空间有理同调与挠同伦群中的指数增长

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

本文利用积分方法推广了自由环路空间有理同调群增长的结果，构造出同伦群$p$-挠指数增长的新空间族，并提出有理同伦群与$p$-挠同伦群指数增长间存在紧密联系的猜想。

On smooth manifolds with homotopy type of a homology sphere

In this paper we study $\mathcal M(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $Ψ$ from $\mathcal M(X)$ into the quotient of $K(X)=[X,BSO]$ by the action of the group of homotopy classes of simple self equivalences of $X$. The map $Ψ$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal M(X)$. We determine the image of $Ψ$ when $X$ belongs to a certain class of homology spheres. In particular, we find conditions on elements of $K(X)$ that guarantee they are pullbacks of normal bundles of manifolds …

同调自接近数与上纤维化序列

Homology self closeness number and cofibration sequence

本文研究了单连通CW复合同调自接近数及同伦上纤维的该数，探讨了同调分解中上纤维化空间自映射，并分析了同调分解与同伦分解间同调自接近数的关系。

Homotopy theory with bornological coarse spaces

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.

Homotopy theory for digraphs

We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs. Note that the homotopy theory for graphs coincides wi…

A Homotopy Theoretic Proof of the BV Identity in Loop Homology

Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological p…

Nonstandard homology theory for uniform spaces

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…

(Co)Homology Self-closeness Numbers of Simply-connected Spaces

The (co)homology self-closeness number of a simply-connected based CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions$\leq k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the mod $p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and stud…

Rational torus-equivariant stable homotopy V: the torsion Adams spectral sequence

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.

Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities

In this paper we study manifolds, $X_Σ$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_Σ)$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_Σ)$ is homotopy invariant under certain surgeries on $X_Σ$.

Hodge Theory for Intersection Space Cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering met…

有向图及其旗复形的同伦概念

A notion of homotopy for directed graphs and their flag complexes

本文为有向图及其旗复形引入同伦等价关系，证明其诱导同调同构，并据此建立过滤有向图持续同调的稳定性定理，同时指出与路径同调的差异。

立方同调的Hurewicz定理

The Hurewicz theorem for cubical homology

本文利用立方集环路空间概念，为立方Kan复形给出了Hurewicz定理的初等证明，揭示了同伦群与同调群之间的关系。

André-Quillen（上）同调与等变稳定同伦论

Andre-Quillen (co)homology and Equivariant Stable Homotopy Theory

针对等变背景下交换化函子类似物的问题，本文基于Hill的真导子等概念扩展到不完全Tambara函子，引入余切复形并构造基本谱序列类似物，同时修正了生成传递长正合序列的假设。

Fibrations up to an equivalence, homotopy colimits and pullbacks

We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence.

有理同伦理论中的同调幂零性：胞腔附着、收缩塔、发散性与稳定化

Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization

该研究在有理同伦理论中构造了nil_h(M)=3<Hnil(M)=4的极小Sullivan代数，揭示同调幂零性的商缺陷可通过逐次有限单连通有理球楔积稳定化消除，建立了稳定化后同调幂零性与同伦范畴的关系。

非阿贝尔同调与类ifying空间图的同调极限

Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups

本文研究了群图的非阿贝尔同调群，证明了其与类ifying空间图的同调极限的同调群同构，并发展了寻找同调极限最小维数非零同调群的方法。

O-minimal homotopy and generalized (co)homology

This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspo…

Natural transformations relating homotopy and singular homology functors

The category of topological spaces endowed with two marked points is equipped with two families $\mathbf F_n$ and $\mathbf H_n$ of functors to the category of abelian groups, indexed by a nonnegative integer $n$: namely, the functor $\mathbf F_n$ takes the object $(X,x,y)$ to the quotient of $\mathbb Zπ_1(X,x,y)$ by an abelian subgroup associated with the $n+1$-st power of the augmentation ideal of the group algebra $\mathbb Zπ_1(X,x)$, and the functor $\mathbf H_n$ takes the same object to the $n$-th singular homology group of $X^n$ relative to a subspace defined in terms of partial diagonals…

Freeness and equivariant stable homotopy

We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, the …

Localisation and Completion with an addendum on the use of Brown-Peterson homology in stable homotopy

These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent w…

On the mod-$\ell$ homology of the classifying space for commutativity

We study the mod-$\ell$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an …

The homotopy fixed points of the circle action on Hochschild homology

We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.

The homotopy invariance of cyclic homology of $A_\infty$-algebras over rings

In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_\infty$-algebra over any field is isomorphic to the cyclic homology of the $A_\infty$-algebra of homologies for the source $A_\infty$-algebra.

CoHochschild homology of chain coalgebras

Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy…

小半群的同伦类型

Homotopy Types of Small Semigroups

提出利用投射分解结构快速计算有限半群同调群的软件包，给出小阶半群和幺半群的计算结果，并构造反例反驳Nico猜想。

On the homology of the double cobar construction of a double suspension

The double cobar construction of a double suspension comes with a Connes-Moscovici structure, that is a homotopy G-algebra (or Gerstenhaber-Voronov algebra) structure together with a particular BV-operator up to a homotopy. We show that the homology of the double cobar construction of a double suspension is a free BV-algebra. In characteristic two, a similar result holds for the underlying $2$-restricted Gerstenhaber algebra. These facts rely on a formality theorem for the double cobar construction of a double suspension.

The homotopy theory of Khovanov homology

We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.

The fiber of persistent homology for trees

Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is $D$. We find that each component is homotopy equivalent to a configuration space on $X$ with specialized constraints encoded by the merge tree. For barcodes $D$ with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.

The homotopy type of the loops on $(n-1)$-connected $(2n+1)$-manifolds

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres

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.

Algebraic structure of the space of homotopy classes of cycles and singular homology

Described the algebraic structure on the space of homotopy classes of cycles with marked topological flags of disks. This space is a non-commutative monoid, with an Abelian quotient corresponding to the group of singular homologies $H_k(M)$. For the marked flag contracted to a point the multiplication becomes commutative and the subgroup of spherical cycles corresponds to the usual homotopy group $π_k(M)$.

The homotopy invariance of dihedral homology of involutive $A_\infty$-algebras over rings

The dihedral homology functor $HD:A_\infty^{\rm inv}(K)\to GrM(K)$ from the category $A_\infty^{\rm inv}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive $A_\infty$-algebras into isomorphisms of graded modules.

任意范畴中的同调与同伦

Homology and homotopy for arbitrary categories

本文通过公理化方法重新建立了任意范畴中同伦与同调的联系，并将凸性重新解释为外在属性。

Persistent homology with non-contractible preimages

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).

On the homology and homotopy of commutative shuffle 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…

Digital homotopy relations and digital homology theories

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 …

超图同调理论的立方体方法

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。

持续同伦群的初等方法

Elementary Methods for Persistent Homotopy Groups

该研究探索持续同伦群的基础性质，推导多个经典同伦定理的持续版本，还通过计算烷烃能量景观的下水平集持续同伦群，证明其能捕捉补充持续同调信息的非平凡特征。

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy c…