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?
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.
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.
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.
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 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.
This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theo…
In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The Main Theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new ine…
We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that…
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar constructions based on "fat" or ribbon graphs, we describe how to construct string topology operations on the loop space of a manifold, using Morse theoretic techniques. Finally, we discuss how to relate …
Milnor metric for Morse--Smale flows from field theory
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；Department of Mathematics, University of Pavia（帕维亚大学数学系）；INFN Sezione di Pavia（INFN帕维亚分部）
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.
We have been interested in understanding the class of 7-dimensional closed and simply-connected manifolds in geometric and constructive ways. We have constructed explicit fold maps, which are higher dimensional versions of Morse functions, on some of the manifolds, previously. The studies have been motivated by studies of {\it special generic} maps, higher dimensional versions of Morse functions on homotopy spheres with exactly two singular points, characterizing them topologically except $4$-dimensional cases. The class contains canonical projections of unit spheres for example. This class ha…
We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $ω$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold $M^n$ with a deformed differential $d_ω=d +λω$. Using Witten's approach to the Morse theory one can estimate the number of critical points of $ω$ in terms of the cohomology of deformed de Rham complex with sufficiently large values of $λ$ (torsion-free Novikov's inequalities). We show that for an interesting class of solvmanifolds this coh…
Let (M,g) be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of points p,q in M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each "generic" initial condition f0, the solution to the Heat Equation is such that for sufficiently large t, f(.,t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.
Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded concentrically. They are, as Morse functions, important in understanding the topologies and the differentiable structures in geometric ways. In the present paper, we study cohomology rings of $3$-dimensional manifolds admitting round fold maps into the plane and see that difference of the coefficient rings and topological types of round fold maps are closely related…
Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof uses homotopy Gerstenhaber algebras in an essential way. I…
This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential cor…
The free loops space $ΛX$ of a space $X$ has become an important object of study particularly in the case when $X$ is a manifold.The study of free loop spaces is motivated in particular by two main examples. The first is their relation to geometrically distinct periodic geodesics on a manifold, originally studied by Gromoll and Meyer in $1969$. More recently the study of string topology and in particular the Chas-Sullivan loop product has been an active area of research. A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of a homogeneo…
As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special generic maps are generalized versions of these maps. Canonical projections of unit spheres are special generic. Studies of Saeki and Sakuma since the 1990s, followed by Nishioka and Wrazidlo, show that the differentiable structures of the spheres and the homology groups of the manifolds (in several classes) are restricted. We see special generic maps are attr…
Morse functions with exactly two singular points on spheres and canonical projections of spheres belong to the class of a certain good class of smooth maps: special generic maps. We mainly investigate information on cohomology of closed and simply-connected manifolds admitting such maps into the $5$-dimensional Euclidean spaces by investigating the embedded curves and submanifolds and their preimages. Studies on homology groups for ones into the Euclidean spaces (whose dimensions are lower than $5$ in most cases) have been pioneered by Saeki and Sakuma since 1990s and later by Nishioka and Wra…
As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental, important and interesting. This paper is on explicit construction of fold maps and homology groups and cohomology rings of their Reeb spaces: they are defined as the spaces of all connected components of preimages of the maps, and in suitable situations inherit some topological information such as homology groups and cohomology rings of the manifolds. Explicit con…
Let $Γ$ be a finite connected graph. The (unlabelled) configuration space $UC^n Γ$ of $n$ points on $Γ$ is the space of $n$-element subsets of $Γ$. The $n$-strand braid group of $Γ$, denoted $B_nΓ$, is the fundamental group of $UC^n Γ$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture…
We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.
We give a proof of a Conjecture of Walker which states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by Farber, Hausmann and Schuetz. In the remaining cases, we use Morse theory and the fundamental group to describe a subring of the cohomology invariant under graded ring isomorphism. From this subring the conjecture can be derived by applying a result of Gubeladze on the isomorphism problem of monoidal rings.
We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.
