arXiv research

Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.

We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.

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.

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…

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.

Study stabilizes Morse-Bott cohomology for equivariant manifolds.

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.

Generalizes Floer homotopy via Morse-Bott theory.

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

Study of circle arrangements related to Morse-Bott functions.

The paper proves manifolds homeomorphic to spheres under specific Morse-Bott conditions.

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…

Constructs a Morse-Bott function on symplectic Grassmannians.

Contractible diffeomorphism groups on lens spaces derived from Morse-Bott foliations.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

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 ho…

Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.

Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.

The study characterizes 3D manifolds using specific Morse-Bott functions.

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 Loja…

We give an alternative proof of that a critical knot of a Morse-Bott function $f: S^3 \rightarrow \mathbb{R}$ is a graph knot where the critical set of $f$ is a link in $S^3$. Our proof inducts on the number of index-1 critical knots of $f$.

Study reconstructs Morse-Bott functions with specific preimage conditions on 3D manifolds.

Develops Bialynicki-Birula and Morse-Bott theory for complex analytic spaces.

Let $f:M \to \mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$…

Research resolves sign conventions in Floer theory for Morse-Bott case.

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$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B…

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, 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 $L^p$-small curvature, when $p>d/2$, to the case of a connection with …

The paper constructs instanton complexes on stratified pseudomanifolds.

The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.

The paper studies diffeomorphisms of a specific foliation on a Klein bottle.

In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spin$^c$ structure isomorphic to its conjugate, we define the counterpart in this…

Analytic realization of Thom-Smale complex for G-manifolds.

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical set. This requires a mix of analytic arguments (establishing properties of the m…

Study SO(3) vortices over orbifold Riemann surfaces and monopoles.

The paper studies Morse theory for Lie algebra actions on Riemannian foliations.

Let $f:M\to\mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set $Σ_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f) = \{h \in \mathcal{D}(M) \mid f\circ h=h \}$ the group of diffeomorphisms of $M$ preserving $f$ and…

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is…

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of t…

Study path spaces and their homology, extending loop products and coproducts.

The study simplifies complex functions on surfaces using a special transformation.

Study spider mechanism configuration spaces using squared distance function.

A new dynamical approach connects resolution cohomology to group representations.

We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow li…

The paper constructs real algebraic functions with specific singularities and preimages.

New theory for Hamiltonian actions on special geometric structures.