arXiv subjects
Publications and source records attributed to Dan Burghelea.
This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons and on the completion of unstable manifolds of a generic smooth vector field. The result is stated in Theorem 1.4.
For a smooth Morse-Smale vector field with Lyapunov constraints (Lyapunov function) one shows how and why the non-triviality of the cohomology, as concluded from its additive structure, detects rest points and the multiplicative structure of the cohomology detects instantons (trajectories between rest points). The same remains true for Lyapunov closed one form, a more general Lyapunov constraint, but in this presentation this fact is discussed only informally. These observations are based on the smooth " manifold with corner structures" of the stable/unstable sets and of the set of trajectories of such vector fields. (This paper is a written version of two talks with the same title given at IMAR Bucharest in November 2023.)
This paper reviews the description of "bar codes" for a continuous real-valued map and explains how to recover the Morse complex of a Morse function from them. In this presentation the bar codes appear as the support of two vector-space valued maps, one defined on the Euclidean plane and the other on the "above diagonal" half plane.
We describe the conjugacy classes of the elements of the free product of two groups and their centralizers and, as a consequence, we correct the calculation of the cyclic and periodic cyclic homology of the group ring of the free product of two groups given in a previous paper.
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.
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-Betti numbers and the Novikov complex associated with a Morse closed 1-form with non-degenerated zeros. Precisely, the sum of the multiplicities of the points in the support of the first configuration which correspond to the integer r equals the r-th Novikov-Betti number and that of the points in the support of second configuration which corresponds to the integer r equals the rank of the boundary map in the Novikov complex. We formulate the basic properties of these configurations, the stability property and the Poincare duality property when the compact ANR is a closed orientable topological manifold, which in full generality will be proven in the second and third part of this work.
For a Morse function on a closed orientable Riemannian manifold one introduces the {\it virtually small spectral package} an analytic object consisting of a finite number of analytic quantities derived from the pair, {\it Riemannian metric, Morse function\} which, in principle, can be calculated. One shows that they determine the {\it Torsion } of the underlying space, a parallel to the result that the dimensions of the spaces of harmonic forms calculate the {\it Euler-Poincaré characteristic} of the underlying space and extends the {\it Poincaré Duality} between harmonic forms and between Betti numbers for a closed oriented Riemannian manifold .
The Witten deformation associated to a Morse function on a closed Riemannian manifold, via Rellich-Kato theorem, relates analytically the spectral package of the Riemannian manifold (eigenvalues and eigenforms) to the Morse complex defined by the pair (Morse function, Riemannian metric) coupled with the "multivariable harmonic oscillators" associated to the critical points of the Morse function. We survey this relation and discuss some implications, including the finite subset of the spectral package referred to as the "virtually small spectral package" .
In this note one shows that the four persistence diagrams and measures defined in [5] are particular cases of the maps and the measures discussed in [2] and [1]
We extend the configurations discussed in Burghelea's book and Burghelea-Haller's paper on topology of angle-valued maps, equivalently the closed, open and closed-open bar codes from real- or angle-valued maps, to topological closed one forms on compact ANRs. As a consequence one provides an extension of the classical Novikov complex associated to a closed smooth one form and a vector field the form is Lyapunov for, to a considerably larger class of situations. We establish strong stability properties and Poincaré duality properties when the underlying space is a closed manifold. Applications to Geometry, Dynamics and Data Analysis are the targets of our research. A different approach towards such bar codes was proposed in Usher-Zhang's work.
In this note we recall the relations between the barcodes in level and sub-level persistence and make precise their relation with the Morse-Novikov complex of a Morse real- or angle-valued map. The results in this papers are implicit in my previous work(w.collaborators), but apparently not well known even to experts.
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 which has the same support as the first configuration. When the field is a the field of complex numbers the second configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of the ANR defined by the angle-valued map. One discusses the properties of these configurations namely, robustness with respect to continuous perturbation of the angle-valued map and the Poincaré Duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov-Betti numbers replacing standard homology and standard Betti numbers.
In this paper one presents a collection of results about the "bar codes" and "Jordan blocks" introduced by Burghelea-Day as "computer friendly" invariants of a tame angle-valued map and one relates these invariants to the Betti numbers, Novikov Betti numbers and the monodromy of the underlying space and map. Among others, one organizes the bar codes as two configurations of points in C\0 and one establishes their main properties: stability property and when the underlying space is a closed topological manifold, Poincaré duality property. One also provides an alternative "computer friendly" definition of the monodromy of an angle valued map based on the algebra of linear relations as well as a refinement of Morse and Morse-Novikov inequalities.
We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of vector spaces indexed by points in complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product these vector spaces are canonically realized as mutually orthogonal subspaces of the homology. The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite dimensional complex vector space. A number of remarkable properties of the above configurations are discussed.
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. Although not discussed in this paper but discussed in works this report is based on, these invariants are computer friendly (i.e. can be calculated by computer implementable algorithms when the source of the map is a simplicial complex and the map is simplicial) and are of relevance for the dynamics of flows which admit Lyapunov real or angle valued map.
In this paper we consider the definition of " monodromy of an angle valued map" based on linear relations as proposed in Burghelea-Haller (3). This definition provides an alternative treatment of monodromy and computationally an alternative calculation of the "Jordan cells", topological persistence invariants of a circle valued maps introduced in Burghelea-Day (2). We give a new geometric proof that the monodromy is actually a homotopy invariant of a pair consisting of a compact ANR and an integral degree one cohomology class without any reference to the infinite cyclic cover associated to cohomology class as in (3), or to the graph representation associated an angle valued map defining the cohomology class as in (2). Most important, we describe an algorithm to calculate the monodromy for a simplicial angle valued map defined on a finite simplicial complex, providing a new algorithm for the calculation of the Jordan cells of the map, shorter than the one proposed in (2). We indicate the computational usefulness of "Jordan cells", and in particular of the proposed algorithm, for the calculation of other basic topological invariants of the pair.
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 refines the Novikov - Betti number and the configuration of free modules refines the Novikov homology associated with the cohomology class defined by f, in the same way the collection of eigenvalues and of generalized eigen-spaces refine the dimension of the vector space and the vector space on which the operator acts. In the case the field is the field of complex numbers the configuration of free modules induces by "von-Neumann completion" a configuration of mutually orthogonal closed Hilbert submodules of the L 2--homology of the infinite cyclic cover of X determined by the map f, which is an Hilbert module over the von-Neumann algebra of complex L-infinity functions on the unit circle in the complex plane.
We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle valued map on an input simplicial complex.