Morse-Bott Functions Research Articles

A connection between cut locus, Thom space and Morse–Bott functions

A connection between cut locus, Thom space and Morse–Bott functions

Realization of a digraph as the Reeb graph of a Morse–Bott function on a given surface

We characterize the set of directed Reeb graphs of Morse–Bott functions on a given surface. Namely, we prove a criterion on realization of a given digraph as the directed Reeb graph of a Morse–Bott function on a given surface.

Realization of a graph as the Reeb graph of a height function on an embedded surface

We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.

Homotopic Distance and Generalized Motion Planning

We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

Witten deformation using Lie groupoids

We express Witten's deformation of Morse functions using deformation to the normal cone and C⁎-modules. This allows us to obtain asymptotics of the ‘large eigenvalues’. This is then applied to the case of Morse-Bott functions inspired by [2].Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack [10] in a slightly more general situation.

The topology of Bott integrable fluids

<p style='text-indent:20px;'>We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.</p>

Topological classification of circle-valued simple Morse-Bott functions on closed orientable surfaces

Topological classification of circle-valued simple Morse-Bott functions on closed orientable surfaces

Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik–Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions.

A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

Abstract We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

English

We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.

Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

Abstract For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

On Morse–Smale–Witten type cohomologies for orbifolds

Let X:=[M/G] be a compact orbifold given by the quotient of a connected compact Lie group G action on an oriented smooth closed manifold M, where the action is assumed to be smooth, effective, locally free and orientation preserving. Let f¯:X→R be a Morse function which lifts to a G-invariant Morse–Bott function f on M. We construct an explicit isomorphism between the Morse–Bott cohomology of f and the orbifold Morse cohomology of f¯ by constructing a new filtered cochain complex out of basic forms to connect the Morse–Bott cochain complex of f and the orbifold Morse–Smale–Witten cochain complex of f¯.

Non-linear Morse–Bott functions on quaternionic Stiefel manifolds

In the Stiefel manifold Xn,k, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n−2k. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.

Diffeomorphisms preserving Morse–Bott functions

Let f:M→R be a Morse–Bott function on a closed manifold M, so the set Σf of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f)={h∈D(M)∣f∘h=h} the group of diffeomorphisms of M preserving f and let D(Σf) be the group of diffeomorphisms of Σf. We prove that the “restriction to Σf” map ρ:S(f)→D(Σf), ρ(h)=h|Σf, is a locally trivial fibration over its image ρ(S(f)).

Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups O(n). We describe the critical loci of the quadratic trace function Tr(AXBXT) and determine their indices via perfect fillings of tables. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod 2 Betti numbers of SO(n).

Morse--Bott Inequalities for Manifolds with Boundary

In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.

Topological classification of circle-valued simple Morse-Bott functions

Topological classification of circle-valued simple Morse-Bott functions

Matrix factorization of Morse–Bott functions

For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse-Bott critical locus $Y\subset X$, Kapustin, Rozansky and Saulina suggest that the associated matrix factorisation category $\mathrm{MF}(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle of $Y$). I confirm their conjecture in the special case when the first neighbourhood of $Y$ in $X$ is split, and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general one-parameter deformation of a $K3$ surface $Y$.

On the Berger conjecture for manifolds all of whose geodesics are closed

We define a Besse manifold as a Riemannian manifold (M, g) all of whose geodesics are closed. A conjecture of Berger states that all prime geodesics have the same length for any simply connected Besse manifold. We firstly show that the energy function on the free loop space of a simply connected Besse manifold is a perfect Morse–Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results, then lead to a solution of Berger’s conjecture when the underlying manifold is a sphere of dimension at least four.