Circle valued Morse theory and Novikov homology

Morse-Novikov cohomology of locally conformally Kähler surfaces

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_θ$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.

Alternative to Morse-Novikov Theory for a closed 1-form (II)

This paper is a continuation of Alternative to Morse-Novikov Theory for a closed 1-form(I), and establishes: a) a refinement of Poincaré duality to an equality between the configurations $^{BM} δ^ω$ and $δ^ω$ resp. $^{BM} γ^ω$ and $γ^ω$ in complementary dimensions, b) the stability property for the configurations $δ^ω_r,$ a) a result needed for the proof of Theorems 1.2 and 1.3 stated the paper mentioned above.

具有微分 graded 系数的 Morse-Novikov 同调的纤维化定理与 Chas-Sullivan 积

A Fibration Theorem and Chas-Sullivan product for Morse-Novikov Homology with differential graded coefficients

本文将具有微分 graded 系数的 Morse 同调及 Chas-Sullivan 积扩展至 Morse-Novikov 同调，构建了相应的 Morse-Novikov 复形，并在纤维化条件下证明了 Novikov 完备化同调上存在 Chas-Sullivan 类型的积。

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,ω)$ with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)}=0$. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic function/hamiltonian. The gluing result is a type of Maye…

On the Morse-Novikov number for 2-knots

Let $K\subset S^4$ be a 2-knot, that is, a smoothly embedded 2-sphere in $S^4$. The Morse-Novikov number $\mathcal M\mathcal N(K)$ is the minimal possible number of critical points of a Morse map $S^4\setminus K\to S^1$ belonging to the canonical class in $H^1(S^4\setminus K)$. We prove that for a classical knot $K\subset S^3$ the Morse-Novikov number of the spun knot $S(K)$ is $\leq 2\mathcal M\mathcal N(K)$. This enables us to compute $\mathcal M\mathcal N(S(K))$ for every classical knot $K$ with tunnel number 1.

Twisted Novikov homology of complex hypersurface complements

We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self-contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov homology groups associated to positive cohomology classes of degree one defined on the complement.

Alternative to Morse-Novikov Theory for closed 1-form (I)

This paper extends the Alternative to Morse-Novikov theory we have proposed in Burghelea (New topological invariants for real- and angle valued maps, World Scientific, Hackensack, 2018) from real- and angle-valued map to closed 1-forms. For a topological closed 1-form on a compact ANR (= absolute neighborhood retract), a concept generalizing closed differential 1-form on a compact manifold, under the mild hypothesis of tameness, a field and a non-negative integer we propose two configurations of points, the first on the real line the second on the positive real line, which recover Novikov-Bett…

New invariants for a real valued and angle valued map (an Alternative to Morse- Novikov theory)

This paper but section 6 is essentially my lecture at The Eighth Congress of Romanian Mathematicians, June 26 - July 1, 2015, Iasi, Romania. The paper summarizes the definitions and the properties of the invariants associated to a real or an angle valued map in the framework of what we call an Alternative to Morse-Novikov theory. These invariants are configurations of points in the complex plane, configurations of vector spaces or modules indexed by complex numbers and collections of Jordan cells. The first are refinements of Betti numbers, the second of homology and the third of monodromy. Al…

Morse-Novikov理论与Floer理论的抽象层间持续性

Abstract interlevel persistence for Morse-Novikov and Floer theory

本研究构建含庞加莱-诺维科夫结构与滤过匹配对的通用代数框架，为Morse函数层间集同调条形码提供抽象方法，可推广至Floer理论，所得条形码满足稳定性与对偶性，在哈密顿Floer理论中可导出类谱范数量。

The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.

Refinement of Novikov - Betti numbers and of Novikov homology provided by an angle valued map

To a pair (X,f), X compact ANR and f a continuous angle valued map defined on X, a fixed field and a nonnegative integer one assigns a finite configuration of complex numbers with multiplicities located in the punctured complex plane and a finite configuration of free modules over the ring of Laurent polynomials (with coefficients in the fixed field) indexed by the same complex numbers. This is done in analogy with the configuration of eigenvalues and of generalized eigenspaces of an invertible linear operator in a finite dimensional complex vector space. The configuration of complex numbers r…

A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)

For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral cohomology class represented by the map. For each degree the first refinement is a configuration of points with multiplicity located in the punctured complex plane whose total cardinality is the Novikov-Betti number of the pair. The second refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and …

On the Conical Novikov Homology

Let $ω$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([ω])=0$. Let $ξ:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat Λ}_ξ$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $ω$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat Λ}_ξ$ by the set of zeroes of $ω$. In this paper we introduce a refinement of this construction. We define a subring $\hatΛ_Γ$ of ${\hat Λ}_ξ$ and show that the Novikov compl…

Circle-valued Morse theory for frame spun knots and surface-links

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. …

Topology Of Real And Angle Valued Maps And Graph Representations (A Brief Survey)

(lecture delivered at the Congress of the Romanian mathematicians, Brasov, June 2011) Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to Morse-Novicov theory for real or angle valued Morse maps. The invariants are "barcodes" and "Jordan cells". From them one can derive all familiar topological invariants which can be derived via Morse-Novikov theory, like the Betti numbers and in the case of angle valued maps also the Novikov Betti numb…

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a strongly Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent, over L_0, to a bounded complex of finitely generated projective L_0-modules, generalising known results for twisted Laurent polynomial rings.

通过离散莫尔斯理论计算图的2粒子无序构形空间的Borel-Moore同调

Borel-Moore Homology of 2-Particle Unordered Configuration Spaces of Graphs via Discrete Morse Theory

本研究通过离散莫尔斯理论定义图的2粒子无序构形空间的开正则胞腔复形上的莫尔斯函数，借助同构关系得出其Borel-Moore同调群由图的第一贝蒂数决定的结论。

Novikov homology, jump loci and Massey products

Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms …

Computing homology and persistent homology using iterated Morse decomposition

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particular, this approach is provably correct in any dimension.

Morse同调中的路径积与微分格系数

The Path-product in Morse Homology with differential graded coefficients

本文通过Morse理论描述流形路径空间的同调中的弦拓扑积，并在Chas-Sullivan环上定义其模结构。

Persistent homology and Floer-Novikov theory

We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prov…

有限单纯复形上余层同调的莫尔斯-迈耶-菲托里斯序列

A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes

该研究针对有限单纯复形上余层同调，构造迈耶-菲托里斯短正合序列并结合离散莫尔斯理论得到莫尔斯链复形的对应序列，建立其与标准序列的拟同构，还涉及范畴论视角的相关探讨。

Lusternik-Schnirelman theory for closed 1-forms

S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.

Graph magnitude homology via algebraic Morse theory

We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3) prove that the icosahedral graph is diagonal, and (4) compute the magnitude homology of cycles. These results answer several questions of Hepworth and Willerton [HW17].

The Morse-Bott inequalities, orientations, and the Thom isomorphism in Morse homology

The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample and clarify the origin of the mistake.

A Discrete Morse Theory for Hypergraphs

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…

Stable foliations and semi-flow Morse homology

In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration using Conley pairs and their pre-images under the time-$T$-map of the heat flow. A crucial step is to contract each Conley pair onto its part in the unstable manifold. To achieve this we construct stable foliations for Conley pairs using the recently found backward $λ$-Lemma [31]. T…

Morse homology: orientation of the moduli space of gradient flow lines, coherence and applications

In this paper, we shall compute the chain complex and the corresponding homology of some Morse function $f$ over integer coefficients. The definition of the correct boundary operator requires a careful construction of moduli space of (pseudo)gradient flow lines orientations. We will then apply this construction in the computation of these homology groups on 4-manifolds.

Morse-Bott homology

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.

Morse Theoretic Templates for High Dimensional Homology Computation

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We demonstrate the effectiveness of templates in two settings: first, by computing the homology of certain cubical complexes which are homotopy equivalent to $\mathbb{S}^d$ for $1\leq d \leq 20$, and second, by computing Conley complexes and connection matrices for a collection of examples arising from a Conley-Morse theory on spaces of braids diagrams.

Discrete Morse Theory for Weighted Simplicial Complexes

In this paper, we study Forman's discrete Morse theory in the context of weighted homology. We develop weighted versions of classical theorems in discrete Morse theory. A key difference in the weighted case is that simplicial collapses do not necessarily preserve weighted homology. We work out some sufficient conditions for collapses to preserve weighted homology, as well as study the effect of elementary removals on weighted homology. An application to sequence analysis is included, where we study the weighted ordered complexes of sequences.

用于计算过滤余层同调的离散莫尔斯理论

Discrete Morse Theory for Computing Filtered Cosheaf Homology

本文将离散莫尔斯理论扩展至固定胞腔复形上胞腔余层过滤的持续同调计算，提出FilteredScythe算法并证明其正确性与复杂度界，为相关计算提供了方法支持。

Fiber of Persistent Homology on Morse functions

Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown to be the same as the orbit of $f$ under pre-composition $ϕ\mapsto f\circ ϕ$ by diffeomorphisms of $M$ which are isotopic to the identity. Consequently we derive topological properties of the fiber $\mathrm{PH}_f^{-1}(D)$: In particular we compute its homotopy type for many compact surfaces $M$. In the $1$-dimensional settings where $M$ is the unit interval o…

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.

Homología de Morse en variedades compactas

Given a compact Riemannian manifold $(M g)$ and Morse function $f:m\to \mathbb{R}$ whose gradient flow satisfies the Morse-Smale condition, (i.e. the stable and unstable manifolds of f intersect transversely) we construct a chain complex called the Morse-Witten Complex. Our goal on this paper is show that the homology of the Morse-Witten complex is isomorphic to the singular homology of $M$.

Three approaches to Morse-Bott homology

In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the function to a Morse function, the second approach is to use moduli spaces of cascades, and the third approach is to use the Morse-Bott multicomplex. With respect to an explicit perturbation (which can be used to derived the Morse-Bott inequalities), the first two approaches yield the same chain complex up to sign. The third approach is fundamentally differe…

A Discrete Morse Theory for Digraphs

Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology are isomorphic for a transitive digraph. We also study the collapses defined by discrete gradient vector fields. Let $G$ be a digraph and $f$ a discrete Morse function. Assume the out-degree and in-degree of any zero-point of $f$ on $G$ are both 1. We prove that the original digraph $G$ and its $\mathcal{M}$-collapse $\tilde{G}$ have the same path homology g…

Heegaard Floer homology and Morse surgery

We establish surgery formulas for filtration of the Heegaard Floer homology associated with p/q surgery on a null-homologous knot K in a three-manifold Y, induced by K_{p/q}. Here K_{p/q} is the core of the attached solid torus (which produces the surgery). This would generalize a result of Ozsvath and Szabo. We will re-prove that surgery on non-trivial knots can not produce S^3, as a corollary of a non-vanishing result for the knot Floer homology of K_{p/q}.

The moduli space of gradient flow lines and Morse homology

We present a set of notes on Morse Homology, which grew out of lectures the first named autor gave at Ludwig-Maximilian University in Munich, Seoul National University, and the University of Augsburg. Although we do not discuss Floer homology in these notes, they are written having Floer homology in mind. Therefore, we use concepts which can be generalized to the semi-infinite dimensional case. In the finite dimensional case, they might not always be the most efficient ones but on the other hand, the finite dimensional case has the advantage that these concepts can be visualized much more easi…

Floer同调：从广义Morse-Smale动力系统到Forman组合向量场

Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields

本文通过计数闭轨道与孤立临界点间的流线，构造广义Morse-Smale系统的Floer型边缘算子，并推广到Forman组合向量场，从而直接恢复光滑与离散结构的$\mathbb{Z}_2$同调。

(Co)Homology of Lie Algebras via Algebraic Morse Theory

E. Sköldberg's Morse Theory from an Algebraic Viewpoint and M. Jöllenbeck's Algebraic Discrete Morse Theory and Applications to Commutative Algebra, which is the algebraic generalization of R. Forman's discrete Morse Theory for Cell Complexes, is discussed in the context of general chain complexes of free modules. Using this, we compute the Chevalley-Eilenberg (co)homology of the Lie algebra of all triangular matrices $\frak{sol}_n$ over $\mathbb{Q}$ or $\mathbb{Z}_p$ for large enough prime $p$. We determine the column and row in the table of $H_k(\frak{sol}_n;\mathbb{Z})$ where the $p$-torsio…

A Weak $\infty$-Functor in Morse Theory

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspe…

有限支配与强$\mathbb{Z}^2$-格rade环上的Novikov同调

Finite domination and Novikov homology over strongly $\mathbb{Z}^2$-graded rings

本文研究了强$\mathbb{Z}^2$-格rade环上链复形的有限支配性质，证明了在张量乘以特定的八重形式幂级数环后，复形成为单形的条件，扩展了Ranicki、Quinn和第一作者关于单变量和双变量 Laurent 多项式环的结果。

不变与共变的或bi-fold Morse同调

Invariant and Coinvariant Morse Homologies for Orbifolds

本文构建了任意紧致有效或bi-fold的不变和共变Morse链复形，证明其同调为或bi-fold不变量，并推测共变复形同调可计算底层拓扑空间的奇异同调，改进了Cho-Hong的构造。

基于组合多向量场的马尔可夫链中Morse分解的持续同调

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。

开放复形的离散Morse理论

Discrete Morse theory for open complexes

针对开放单纯复形K=X\T，通过阶复形S_K上的离散Morse函数研究K的拓扑变化，并证明K的Borel-Moore同调与梯度向量场诱导的同调同构，推导弱Morse不等式。

扭曲Morse复形

Twisted Morse complexes

本文研究闭有限维光滑流形上的扭曲Morse同调与上同调，证明相关定理与多类同调/上同调的同构关系，并通过多个计算实例验证了扭曲Morse复形的有效性。

主丛的传输函数与微分分次系数的Morse同调

Transport functions for principal bundles and Morse homology with differential graded coefficients

本文用Morse理论描述主丛，通过断裂梯度流线的传输函数编码转移函数，并基于此构造微分分次系数的Morse链复形，证明其同调为伴随丛的同调。

带DG系数的Morse同调中的Goresky-Hingston余积

The Goresky-Hingston Coproduct in Morse Homology with DG Coefficients

本文通过拟同构描述自由环路空间上的Goresky-Hingston余积，并计算了形如$S^n/G$流形的实系数余积，附录提供所需$\nathcal{A}_\infty$-模态射求逆结果。

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examp…

具有球锥奇点的伪流形上的奇异Morse-Smale流：Conley理论与交合同调

Singular Morse-Smale Flows on Pseudomanifolds with Spherical-Cone Singularities: Conley Theory and Intersection Homology

将Morse-Conley理论推广到具有球锥奇点的伪流形，通过引入奇异Morse-Smale流，建立了奇点Conley指数的局部动力学公式，证明了全局Lyapunov函数的存在性，并利用磨光化方法关联奇异与光滑情形，最后通过动力系统方法计算交合同调。

德拉尼复形中的包裹环：架起持续同调与离散莫尔斯理论的桥梁

Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

该研究在形状重建语境下探究持续同调与离散莫尔斯理论的关联，证明Delaunay复形中任意环的字典序最优同调环支撑于同参数Wrap复形，通过环约简与梯度流的基础关联推导结论，打通两类方法的联系。

Circle-valued Morse theory for complex hyperplane arrangements

Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R.

Morse 同调中链级模型的 Chas-Sullivan 乘积研究

A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients

本文提出了一种基于 Morse 理论的链级模型，用于研究自由循环空间的 Chas-Sullivan 乘积，并构建了微分格版本的 Künneth 公式和 Pontryagin-Thom 构造。