Jack Morava
We study the Madsen-Tillmann spectrum $\C P^\infty_{-1}$ as a quotient of the Mahowald pro-object $\C P^{\infty}_{-\infty}$, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.
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Algebraic string topology from the neighborhood of infinity
本文在配备pre-Calabi-Yau结构的光滑$A_\infty$-范畴的Hochschild链复形上构造了Goresky-Hingston环路余积的代数类似物，并利用范畴化形式穿孔无穷邻域和图形形式化给出显式模型，且在球面情形完成了显式计算。
We give an introduction for the non-expert to TQFT (Topological Quantum Field Theory), focussing especially on its role in algebraic topology. We compare the Atiyah axioms for TQFT with the Eilenberg Steenrod axioms for homology, give a few simple examples of TQFTs, and discuss some other approaches that have been taken to defining TQFT. We then propose a new formulation of TQFT, which is closer in spirit to the way conventional functors of algebraic topology, like homology, are presented. In this approach the fundamental operation of gluing is incorporated through the notion of a *gluing morp…
A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.
String topology in three flavours
本文从几何与代数两个视角研究Chas-Sullivan积和Goresky-Hingston余积，计算透镜空间上的运算并证明余积可区分三维透镜空间，同时用有理同伦论建立两种定义的等价性并讨论不变性。
This article contains a collection of problems contributed during the course of the conference.
The efficiency of contemporary algebraic topology is not optimal since the category of topological spaces can be made more algebraic by introducing a profoundly new (-1)-dimensional topological space as a topological join unit. Thereby synchronizing the category of topological spaces with the structures within the contemporary category of simplicial complexes as well as with the structures within the algebraic categories. In the category of topological spaces, the empty space Ø has since long been given the role as a join unit - ad-hoc though. Since it is {Ø}, not Ø, that is the join unit with…
This is a slightly edited version of the transparencies for a seminar at UCL, May 7, 2003. It is intended to give a quick view of background, ideas, and some calculations, in the applicatioon of some non commutative methods to algebraic topology.
The classical ``computation'' methods in Algebraic Topology most often work by means of highly infinite objects and in fact +are_not+ constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program +Kenzo+ has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenz…
We observe that the oriented homeomorphism type of a simply- connected smooth projective surface is determined by its algebraic structure modulo an odd prime of good reduction.
We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be developed as an enriched version of directed algebraic topology, where illicit paths are penalised with an infinite cost, and the licit ones are measured. Its algebraic counterpart will be 'weighted algebraic structures', equipped with a sort of directed seminorm. In the fundamental weighted category of a generalised metric space, introduced here, each homotopy…
The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.
We show in this text how the most important homology equivalences of fundamental Algebraic Topology can be obtained as reductions associated to discrete vector fields. Mainly the homology equivalences whose existence -- most often non-constructive -- is proved by the main spectral sequences, the Serre and Eilenberg-Moore spectral sequences. On the contrary, the constructive existence is here systematically looked for and obtained.
Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new…
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and invariants, characteristic classes). Thus main ideas of algebraic topology are presented with minimal technicalities. Familiarity of a reader with basic notions of topology (such as 2-dimensional manifolds and vector fields) is desirable, although definitions are given at the beginning. The book is accessible to undergraduates and could also be an interesting easy rea…
This talk gave a sketch of the contents and background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker and Rafael Sivera (Valencia). The aim is to give in one place a full account of work by R. Brown and P.J. Higgins since the 1970s which defines and applies crossed complexes and cubical higher homotopy groupoids in algebraic topology and group cohomology.
We give a new approach to the study of statistical mechanical systems: algebraic topology is used to investigate the statistical distributions of stochastic currents generated in graphs. In the adiabatic and low temperature limits we will demonstrate that quantization of current generation occurs.
Algebraic Topology of Certain Sasaki Joins
本文研究正亏格曲面圆丛与加权三维球面形成的Sasaki连接的代数拓扑，计算其基本群、同调与上同调不变量，并识别由挠产生的链环形式。
We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.
We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to Ω(S^1 \wedge X)\to Ω(S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X = (S^1)^m \wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)} \otimes π_{n+1+i}^{A1} (S^{2n+2m+1} \wedge (G_m)^{\wedge 2q}) \Rightarrow Z_{(2)} \otimes π_{i}^{A1, s} (S^{m} \wedge (G_m)^{\wedge q}).$$
In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.
We survey the topology which led to the original bar and cobar constructions, for both associative algebras and coalgebras and for Lie algebras and commutative coalgebras. These constructions are often viewed as part of the larger theory of Koszul duality of operads, so this survey is meant to offer an historical perspective on the most prominent cases of that theory. We also explain recent work which shows that Hopf/linking invariants for homotopy are at the heart of the duality between commutative algebras and Lie coalgebras.
Weighted algebraic topology, II (Real valued metrics)
本文研究实值度量，扩展了Lawvere的度量空间概念，探讨了如何用实数线测量过程的盈亏，引入了线性实值度量与势函数的关系，为加权代数拓扑提供理论基础。
On the topology of real algebraic stacks
本文发展多种方法研究实Deligne-Mumford栈实轨迹的拓扑，并验证了若干情形下Smith-Thom型不等式，结合了群论、代数几何与拓扑学技术。
We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [5, 8], which we extend to a similar construction of a Dijkgraaf- Witten type TQFT in the case where the defects (lower dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orien…
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the integral cohomology of the ordered---and the mod $p$ cohomology of the unordered---configuration spaces of $\mathbb{R}^n$, and the rational cohomology of the unordered configuration spaces of an arbitrary manifold of odd dimension. We also discuss models for mapping spaces in terms of labeled configuration spaces, and we show that these models split stably. Som…
The main aim of this paper is to explore the ideas of persistent homology and extended persistent homology, and their stability theorems, using ideas from [Bubenik and Scott, 2014; Cohen-Steiner, Edelsbrunner, and Harer, 2007; and Cohen-Steiner, Edelsbrunner, and Harer, 2009], as well as other sources. The secondary aim is to explore the homology (and cohomology) of non-orientable surfaces, using the Klein bottle as an example. We also use the Klein bottle as an example for the computation of (extended) persistent homology, referring to it throughout the paper.
Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $Mξ$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a grou…
In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of "toric topology" we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf-Ness sets. In the case …
We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.
We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.
In the 70:th, combinatorialists begun to systematically relate simplicial complexes and polynomial algebras, named Stanley-Reisner rings or face rings. This demanded an algebraization of the simplicial complexes, that turned the empty simplicial complex into a zero object w.r.t. to simplicial join, losing its former role as join-unit - a role taken over by a new (-1)-dimensional simplicial complex containing only the empty simplex. There can be no realization functor targeting the classical category of topological spaces that turns the contemporary simplicial join into topological join unless …
Algebraic topology of $C^*$-algebras
本文研究$C^*$-代数的Gelfand空间，将其作为拓扑空间的推广，探讨其在非交换$C^*$-代数中的应用，提出若干经典代数拓扑概念的$C^*$-代数类比。
In the 1990's, Itenberg and Haas studied the relations between combinatorial data in Viro's patchworking and the topology of the resulting non-singular real algebraic curves in the projective plane. Using recent results from Renaudineau and Shaw on real algebraic curves near the non-singular tropical limit, we continue the study of Itenberg and Haas inside any non-singular projective toric surface. We give new Haas' like criteria for patchworking non-singular real algebraic curves with prescribed number of connected components, in terms of twisted edges on a non-singular tropical curve. We the…
There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimensional 3D fluid algebra", write its Euler equation and derive properties related to energy, helicity, transport of vorticity and linking that characterize this equation. This is directly motivated by the infinite dimensional fluid algebra associated to a closed riemannian three manif…
Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces
本研究基于网理论探讨收敛空间，简化连续收敛描述并应用于同伦理论，在极限空间中定义基本广群并证明了 Seifert-van Kampen 定理的广群版本。
After two papers on weak cubical categories and {\it collarable} cospans, respectively, we put things together and construct a {\it weak} cubical category of cubical {\it collared} cospans of topological spaces. We also build a second structure, called a {\it quasi} cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has {\it lax} identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree.
In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 manifolds with special properties, which…
Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain…
Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).
We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type…
Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula
本文利用机器学习探究球面同伦群生成元的群论结构，在Wu公式的单纯群设定下，将生成单纯循环转化为从Dyck语言相关数据集交集中采样的问题，并提出并评估了采用多标记信息的语言建模方法及非神经基线。
Algebraic Topology for Data Scientists
本书全面介绍了拓扑数据分析（TDA），旨在帮助缺乏背景的读者掌握点集拓扑、抽象代数和同调理论等基础知识，并探讨TDA的应用及上同调等高级主题。
Capturing Tie Strength with Algebraic Topology
本文利用代数拓扑提出三种度量高阶网络中连接强度的指标，并构建一种综合中心性度量，在15个数据集上优于标准代理指标，揭示了弱连接的信息优势。
The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most expressive and generalizable architectures suited to that data, beyond inductive bias. After suggesting algebraic topology as a measure for data complexity, we show that the power of a network to express the topological complexity of a dataset in its decision region is a strictly limiting factor in its ability to generalize. We then provide the first empirical cha…
A new quantum computational set-up for algebraic topology via simplicial sets
本文提出一种基于单纯集合理论的代数拓扑量子计算框架，通过构建有限维单纯Hilbert空间解决同调计算问题，并设计了在有限资源下计算同调空间与Betti数的量子算法方案。
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