Complex Homotopy Research Articles

The covering type of closed surfaces and minimal triangulations

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent to X. In this article we compute the covering type of all closed surfaces. Our results completely settle a problem posed by Karoubi and Weibel, and shed more light on the relationship between the topology of surfaces and the number of vertices of minimal triangulations from a homotopy point of view.

Classification and characterization of rationally elliptic manifolds in low dimensions

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.

Arboreal singularities

We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.

Coxeter transformation groups and reflection arrangements in smooth manifolds

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group on a real vector space $V$. The group acts properly and fixes a union of hyperplanes. The $W$-action extends as the covering space action to the complexified complement of these hyperplanes. The fundamental groups of the complement and the orbit space are the pure Artin group and the Artin group respectively. For the Artin groups of finite type Deligne proved that the associated complement is aspherical. Using the Coxeter group data Salvetti gave a construction of a cell complex which is a $W$-equivariant deformation retract of the complement. This construction was independently generalized by Charney and Davis to the Artin groups of infinite type. A lot of algebraic properties of these groups were discovered using combinatorial and topological properties of this cell complex. In this paper we represent a Coxeter group as a subgroup of diffeomorphisms of a smooth manifold. These so-called Coxeter transformation groups fix a union of codimension-$1$ (reflecting) submanifolds and permute the connected components of the complement. Their action naturally extends to the tangent bundle of the ambient manifold and fixes the union of tangent bundles of these reflecting submanifolds. Fundamental group of the tangent bundle complement and that of its quotient serve as the analogue of pure Artin group and Artin group respectively. The main aim of this paper is to prove Salvetti's theorems in this context. We show that the combinatorial data of the Coxeter transformation group can be used to construct a cell complex homotopy equivalent to the tangent bundle complement and that this homotopy equivalence is equivariant.

A complex line homotopy algorithm for coupled algebraic Riccati equations

This paper presents an algorithm based on a complex line homotopy mapping for the numerical solution of systems of coupled algebraic matrix Riccati equations which often met in continuous time control system design problems. Well-defined complex homotopy paths lead to solutions globally from independent initial conditions, and they are computed via continuous dynamic flow algorithms (ordinary differential equation solvers). Although the homotopy mapping is complex it is shown how it is possible in practice to obtain desired real solutions of certain definiteness even from zero initial conditions. Moreover, a scaling method is deployed within the algorithms for numerical enhancement and its effect to homotopy paths, convergence and stability is experimentally studied. In addition, a first-order perturbation analysis framework for the assessment of computed solutions is provided. Comparisons with other existing methods from the literature are made through several numerical examples considering mixed H|$_{2}$|/H|$_{\infty }$|⁠, Nash games and stochastic control system design problems, where the efficiency of the proposed algorithm is illustrated. It is worth noticing that algorithmic implementation and numerical experimentation with complex homotopies for the coupled algebraic Riccati equations under study appears to be absent the literature. Hence, the contribution of the present paper to related problems in systems and control.

Sparsity‐based space–time adaptive processing using complex‐valued Homotopy technique for airborne radar

In this study, a novel sparsity-based space–time adaptive processing algorithm based on the complex-valued Homotopy technique is proposed for airborne radar applications. The proposed algorithm firstly extends the existing standard real-valued Homotopy method to a more general complex-valued application using the gradient approaches. By exploiting the sparsity of the clutter spectrum in the whole spatiotemporal plane, the proposed algorithm recovers the clutter spectrum via the proposed complex Homotopy algorithm and then uses it to estimate the clutter covariance matrix, followed by the space–time filtering and the target detection. Furthermore, the implementations of the proposed algorithm are detailed. The computational complexity analysis shows that the proposed algorithm has a lower-computational complexity than the existing complex-valued Homotopy algorithm. Simulation results show that the proposed algorithm converges at a very fast speed (only 4–6 snapshots in the authors simulations) and provides both excellent detection performance and easy parameter settings.

Complex homotopy and Grunsky operator

We establish the underlying quasiconformal features of the generalized Grunsky coefficients and provide their applications to qualitative and quantitative investigations of the complex holomorphic isotopies.

Sparse channel estimation for wavelet-based underwater acoustic communications

In this paper, concerning underwater acoustic (UWA) communications, we propose a wavelet filter bank system as the extension of orthogonal frequency-division multiplexing system. We exploit the convolutional structure of a UWA channel and formulate the pilot-assisted channel estimation as a sparse recovery problem. Then, we investigate the restricted isometry property of the measurement matrix via eigenvalue analysis and Gershgorin circle theorem. The sparse recovery problem is proven to satisfy the restricted isometry property. Moreover, we also propose a low-complexity complex-field homotopy algorithm for sparse channel estimation, regarding the fact that the channel taps of each path are usually complex valued in practice. Simulation results show that the wavelet filter bank system achieves more accurate UWA channel estimation performance than orthogonal frequency-division multiplexing system under the same conditions of bandwidth, duration, data rate and channel profile. The proposed complex homotopy algorithm outperforms orthogonal matching pursuit (OMP) and stagewise OMP in both systems, whereas its computational complexity is similar to OMP. Copyright © 2012 John Wiley & Sons, Ltd.

Application of numerical continuation to compute all solutions of semilinear elliptic equations

Abstract We adapt numerical continuation methods to compute all solutions of finite difference discretizations of nonlinear boundary value problems involving the Laplacian in two dimensions. New solutions on finer meshes are obtained from solutions on coarser meshes using a complex homotopy deformation. Two difficulties arise. First, the number of solutions typically grows with the number of mesh points and some form of filtering becomes necessary. Secondly, bifurcations may occur along homotopy paths of solutions and efficient methods to swap branches are developed when the mappings are analytic. For polynomial nonlinearities we generalize an earlier strategy for finding all solutions of two-point boundary value problems in one dimension and then introduce exclusion algorithms to extend the method to general nonlinearities.

Guaranteeing the Homotopy Type of a Set Defined by Non-Linear Inequalities

This paper provides an effective method to create an abstract simplicial complex homotopy equivalent to a given set \(\mathbb{S}\) described by non-linear inequalities (polynomial or not). To our knowledge, no other numerical algorithm is able to deal with this type of problem. The proposed approach divides \(\mathbb{S}\) into subsets that have been proven to be contractible using interval arithmetic. The method is close to Cech cohomology and uses the nerve theorem. Some examples illustrate the principle of the approach. This algorithm has been implemented.

Floer homology and homotopy planes

We compute the Floer homology of a certain family of homology 3spheres occuring as the boundary of a compactification of complex homotopy planes. The Floer complex turns out to be concentrated in even degrees and 2-periodic. This is proved by using suitable bordisms to Seifert fibred spaces.

On local systems over complements to arrangements of hyperplanes associated to grassmann strata

In the first two parts we recall the construction of generalized hypergeometric functions and of the cellular complex homotopy equivalent to the complement of a family of hyperplanes in CN. In the third part we find a generalization of some results in [2] about the homology of local systems on an affine space less some hyperplanes. Our method is based on [7] and it gives also informations about the cellular complex there constructed. In the last part explicit bases for the only non-vanishing homology group are described in terms of the cells of the above mentioned complex. The configurations of hyperplanes which we examine are those giving fundamental strata in the grassmanian ([2], [3]) and strata in G3,n allowing also triple points.

Nonlocal automated sensitivity analysis

Previously a complete ordinary differential equation (ODE) system was developed for tracking the solution x(α) of a parameterized system of nonlinear equations 0 = Ψ( x, α) over an α-interval [α 0, α 1]. This paper develops a two-phase complex homotopy continuation method for obtaining the required initial conditions at α 0. An initial “short” artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function Ψ, which can ameliorate the problem of artificially induced singularities. Also, the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short (minimal number of integration steps) and numerically stable (avoidance of singular points). A FORTRAN program Nasa has been developed for solving the complete ODE system, starting with a two-phase initialization. The program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives. Numerical experiments are reported which illustrate the program's effectiveness.

Singular points of complex surfaces and homotopy

Singular points of complex surfaces and homotopy

Dolbeault homotopy theory

For complex manifolds, we define “complex homotopy groups” in terms of the Dolbeault complex. Many theorems of classical homotopy theory are reflected in the properties of complex homotopy groups. Analytic fibre bundles yield long exact sequences of complex homotopy groups and various Hurewicz theorems relate complex homotopy groups to the Dolbeault cohomology. In a more analytic vein, the classical Fröhlicher spectral sequence has a complex homotopy analogue. We compute these complex homotopy invariants for such examples as Calabi-Eckmann manifolds, Stein manifolds, and complete intersections.