Maciej Starostka
Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.
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For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known~\cite{abbondandoloschwarz, aizenbudzapolski,audindamian, kronheimermrowka, schwarz}, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of iso…
Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell attachment theorem in Morse theory [13] and, secondly, some ideas [12] how to try to organize the closures of the unstable manifolds of a Morse-Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use t…
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…
Let $M^n$ be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. For any $[θ] \in H^1_{dR}(M^n), [θ] \neq 0$, we show that the Morse- Novikov cohomology group $H^p(M^n, θ)$ vanishes for any $p$. A similar result holds for a closed manifold of almost nonnegative Ricci curvature under the additional assumption that its curvature operator is uniformly bounded from below.
In this paper we find sufficient conditions for the vanishing of the Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian manifolds. Several examples are presented, along with applications in the context of l.c.s. and l.c.K. foliations.
We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we consider the relations between Morse-Novikov cohomology and Dolbeault-Morse-Novikov cohomology, moreover, investigate stabilities of their dimensions under the deformations of complex structures. In some aspects, Morse-Novikov and Dolbeault-Morse-Novikov cohomology behave similarly with…
Morse-Novikov cohomology and rigidity of Lie affine foliations
研究二维非阿贝尔李代数仿射(1)建模的李叶状结构的刚性，通过Morse-Novikov上同调给出消失条件，证明3或4维紧致可定向流形上的李仿射叶状结构在形变下是刚性的。
The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_ω=d+ω\wedge$, where $ω$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the…
We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.
Dolbeault-Morse-Novikov Cohomology on Complex manifolds and its applications
本文通过Dolbeault-Morse-Novikov上同调研究复流形拓扑，建立积分不等式，证明在特定条件下Hirzebruch χ_y-亏格消失，并将经典定理推广至复流形。
Inspired by the recent works of S. Rao--S. Yang--X.-D. Yang and L. Meng on the blow-up formulae for de Rham and Morse--Novikov cohomology groups, we give a new simple proof of the blow-up formula for Morse--Novikov cohomology by introducing the relative Morse--Novikov cohomology group via sheaf cohomology theory and presenting the explicit isomorphism therein.
We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward'' dualities are a…
This article arose from a series of three lectures given at the Banach Center, Warsaw, during period of 24 March to 13 April, 2003. Morse functions are useful tool in revealing the geometric formation of its domain manifolds $M$. They define the handle decompositions of $M$ from which the additive homologies $H_{\ast}(M)$ may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative cohomology rather than the additive homology?
Witten instanton complex and Morse-Bott inequalities on stratified pseudomanifolds
本文在分层伪流形上构造Witten瞬子复形，推广Morse-Bott条件并证明相应的Morse不等式，为奇异空间上的几何分析提供新工具。
The weight $θ$-sheaf $\underline{\mathbb{R}}_{X,θ}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of Künneth and Leray-Hirsch types. As applications, we prove that the $θ$-Lefschetz number is independent of $θ$ and calculate the Morse-Novikov cohomologies of projective bundles. Based on these results, we give two blow-up formulae on (\emph{not necessarily compact}) complex manifolds, where the self-intersection formulae play a key role in establishing the explicit expressions for them.
We study the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ of the deRham complex $Λ^*(G/Γ)\otimes{\mathbb C}$ of a compact solvmanifold $G/Γ$ with a deformed differential $d_{λω}=d + λω$, where $ω$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with a completely solvable Lie algebra $\mathfrak{g}$ and a cocompact lattice $Γ\subset G$ the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ coincides with the cohomology $H^*_{λω}(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ associated with the one-dimensional representation $ρ_{λω}: …
In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identically zero.
We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods $\langle \int_γη\mid [γ] \in π_1(M)\rangle$ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for the Inoue surface $\mathcal{S}^0$.
We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.
We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to satisfying the standard L^2 Morse inequalities, they also satisfy certain asymptotic L^2 Morse inequalities. These reduce to the standard Morse inequalities in the finite dimensional case, and when the Morse 1-form is exact. We define the extended twisted L^2 de Rham cohomology and …
The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms…
Equivariant Morse-Bott cohomology through stabilization
本文通过稳定化技术研究等价不变的莫尔斯-博特上同调，证明小扰动产生稳定不变函数，并利用通用度量实现横截性和定向性假设。
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.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse-Novikov class and the Bott-Chern class of a Vaisman ma…
It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. I…
Mapping Cone and Morse Theory
该研究为光滑流形上闭微分形式关联的映射锥复形构建了全新的莫尔斯理论描述，证明其锥莫尔斯上同调与映射锥上同调同构，得到相关维数严格不等式及临界点数与贝蒂数之差的界，适用范围广泛。
We study the classical action functional $\SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold $M$ and the symplectic action $\AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in $M$. The potential $V\in C^\infty(M\times S^1,\R)$ serves as perturbation and we show that both functionals are Morse for generic $V$. In this case we prove that the Morse index of a critical point $x$ of $\SMC_V$ equals minus its Conley-Zehnder index when viewed as a critical point …
We develop an analytic framework for Lefschetz fixed point theory and Morse theory for Hilbert complexes on stratified pseudomanifolds. We develop formulas for both global and local Lefschetz numbers and Morse, Poincaré polynomials as (polynomial) supertraces over cohomology groups of Hilbert complexes, developing techniques for relating local and global quantities using heat kernel and Witten deformation based methods. We focus on the case where the metric is wedge and the Hilbert complex is associated to a Dirac-type operator and satisfies the Witt condition, constructing Lefschetz versions …
We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs $(A,Φ)$, where $A$ is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and $Φ$ is a holomorphic section of $(E, d_A")$. We prove that a certain explicitly defined substratification of the Morse stratification is perfect in the sense of $\G$-equivariant cohomology, where $\G$ denotes the unitary gauge group. As a consequence, Kirwan surjectivity holds for pairs. It also follows that the twist embedding into higher degree induces a surjection on equivariant cohomology. This …
A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds.…
Let $Γ$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $Γ$ is defined by a map, $α$, which assigns to each oriented edge e of $Γ$ a one-dimensional representation of G (or, alternatively, a weight, $α_e$, in the weight lattice of G). For the assignment, $e \to α_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(Γ, α)$ an equivariant cohomology ring, $H_G(Γ)=H(Γ,α)$. By definition this ring contains the equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, …
In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivaria…
We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singular homology of the loop space. In an appendix we perturb the equations in order to eliminate degeneracies and to get to a situation with nondegenerate critical points only. The (co)homology groups turn out to be invariant under the perturbation.
A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow
本文通过动态框架和规范论，构建了莫科雷对应关系中上同调与不可约表示之间的映射，并在循环情况下验证了猜想。
The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-Hölder filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local …
This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal.
Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomol…
Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kähler wedge metrics
本文在带楔 Kähler 度量的分层伪流形上构造 Witten 瞬子复形，将全纯 Morse 不等式推广到奇异空间，并证明相关 $L^2$ 上同调与算子复形的刚性。
Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of index $1$ and $2$ critical points of Morse closed $1$-forms are obtained, that are different in nature from those derived from the Novikov homology.
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。
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