Deformation Retraction Research Articles

Metric thickenings of Euclidean submanifolds

Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris–Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when Y is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the Vietoris–Rips and Čech thickenings, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology. We show that for Euclidean subsets X with positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris–Rips and Čech thickenings of X are homotopy equivalent to X for scale parameters less than the reach). To our knowledge this is the first version of Hausmann's theorem for Vietoris–Rips constructions on entire Euclidean submanifolds (as opposed to Riemannian manifolds), and our result also extends to non-manifold shapes (as not all sets of positive reach are manifolds). In contrast to Hausmann's original proof, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.

On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$ C n

We present a construction of regular Stein neighborhoods of a union of maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in $\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are sufficiently small. Our proof is based on a local construction of a suitable plurisubharmonic function $\rho$ near the origin, such that the sublevel sets of $\rho$ are strongly pseudoconvex and admit strong deformation retraction to $M\cup N$. We also give the application of this result to totally real immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely many double points, and such that the union of the tangent spaces at each intersection in some local coordinates coincides with $M\cup N$, described above.

Local and end deformation theorems for uniform embeddings

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behavior is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ≤0 the κ-cone ends over any compact Lipschitz metric manifolds, all of them have the property (LD). We also formulate a notion of end deformation property for uniform embeddings over proper product ends (ED). For example, the 0-cone end over a compact metric manifold has the property (ED) if it has the property (LD). It is shown that if a metric manifold M has finitely many proper product ends with the property (ED), then the group of bounded uniform homeomorphisms of M endowed with the uniform topology admits a strong deformation retraction onto the subgroup of bounded uniform homeomorphisms which are identity over those ends. We also study a role of uniform isotopies in deformation of uniform homeomorphisms and show that Alexander isotopies in κ-cones induce contractions of some subgroups of groups of bounded uniform homeomorphisms.

Eye retraction and rotation during Corvis ST 'air puff' intraocular pressure measurement and its quantitative analysis.

The aim of this study was to analyse the indentation and deformation of the corneal surface, as well as eye retraction, which occur during air puff intraocular pressure (IOP) measurement. A group of 10 subjects was examined using a non-contact Corvis ST tonometer, which records image sequences of corneas deformed by an air puff. Obtained images were processed numerically in order to extract information about corneal deformation, indentation and eyeball retraction. The time dependency of the apex deformation/eye retraction ratio and the curve of dependency between apex indentation and eye retraction take characteristic shapes for individual subjects. It was noticed that the eye globes tend to rotate towards the nose in response to the air blast during measurement. This means that the eye globe not only displaces but also rotates during retraction. Some new parameters describing the shape of this curve are introduced. Our data show that intraocular pressure and amplitude of corneal indentation are inversely related (r8 =-0.83, P=0.0029), but the correlation between intraocular pressure and amplitude of eye retraction is low and not significant (r8 =-0.24, P=0.51). The curves describing corneal behaviour during air puff tonometry were determined and show that the eye globe rotates towards the nose during measurement. In addition, eye retraction amplitudes may be related to elastic or viscoelastic properties of deeper structures in the eye or behind the eye and this should be further investigated. Many of the proposed new parameters present comparable or even higher repeatability than the standard parameters provided by the CorvisST.

On regular Stein neighborhoods of a union of two totally real planes in $\mathbb {C}^2$

In this paper we find regular Stein neighborhoods for a union of totally real planes $M=(A+iI)\mathbb{R}^2$ and $N=\mathbb{R}^2$ in $\mathbb{C}^2$ provided that the entries of a real $2 \times 2$ matrix $A$ are sufficiently small. A key step in our proof is a local construction of a suitable function $\rho$ near the origin. The sublevel sets of $\rho$ are strongly Levi pseudoconvex and admit strong deformation retraction to $M\cup N$.

Tropical geometry of moduli spaces of weighted stable curves

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces.

The topology of nilpotent representations in reductive groups and their maximal compact subgroups

Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group \Gamma generated by r elements, we consider the representation spaces Hom(\Gamma,G) and Hom(\Gamma,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to prove that there is a strong deformation retraction of Hom(\Gamma,G) onto Hom(\Gamma,K). We also obtain a strong deformation retraction of the geometric invariant theory quotient Hom(\Gamma,G)//G onto Hom(\Gamma,K)/K.

Finite morphisms between projective varieties and Skeleta

In this paper we study finite morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let $V'$ and $V$ be irreducible, projective varieties over an algebraically closed, non- archimedean valued field $k$ and $\phi$ be a finite morphism $\phi : V' \to V$. Let $x \in V^{an}(L)$, where $L/k$ is an algebraically closed complete non-archimedean valued field extension. We associate canonically to $x$ an $L$-point of the space $(V \times_k L)^{an}$ which lies on the fiber over $x$ and denote this point $x_L$. The embedding of $V$ into some $n$-dimensional projective space defines in a natural way a family of open neighbourhoods $\mathcal{O}_{x_L}$ in $(V \times_k L)^{an}$ of $x_L$. Each element of this family is parametrized by an $(n + 1)^2$-tuple which quantifies its size. Of particular interest to us will be those elements $O$ of the set $\mathcal{O}_{x_L}$ whose preimage for the morphism $(\phi \times id_L)^{an}$ decomposes into the disjoint union of homeomorphic copies of $O$ via $(\phi \times id_L)^{an}$. Let $\mathcal{G}_{x_L} \subset \mathcal{O}_{x_L}$ denote the sub collection of elements of this form. Theorem 1.3 shows that there exists a deformation retraction of the space $V$ onto a finite simplicial complex such that along the fibers of the retraction the size of the largest element belonging to $\mathcal{G}_{x_L}$ is constant.

Character varieties of virtually nilpotent Kähler groups and G–Higgs bundles

Let G be a connected complex reductive affine algebraic group, and let K be a maximal compact subgroup. Let X be a compact connected K\"ahler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety Hom(Gamma, G)/G admits a natural strong deformation retraction to the subset Hom(Gamma, K)/K. The natural action of C^* on the moduli space of G-Higgs bundles over X extends to an action of C. This produces the deformation retraction.

Balancing conditions in global tropical geometry

We study tropical geometry in the global setting using Berkovich's deformation retraction. We state and prove the generalized balancing conditions in this setting. Starting with a strictly semi-stable formal scheme, we calculate certain sheaves of vanishing cycles using analytic \'etale cohomology, then we interpret the tropical weights via these cycles. We obtain the balancing condition for tropical curves on the skeleton associated to the formal scheme in terms of the intersection theory on the special fiber. Our approach works over any complete discrete valuation field.

Higgs bundles and representation spaces associated to morphisms

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\,\subset\, G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f\colon X\to Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_*\colon \pi_1(X)\to\pi_1(Y)$ is surjective. Define \begin{align*} {\mathcal R }^f\big(\pi_1(X), G\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), G\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,,\\[6pt] {\mathcal R }^f\big(\pi_1(X), K\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), K\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,, \end{align*} where $A\colon G\to \operatorname{GL}(\operatorname{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(\pi_1(X, x_0),\, G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(\pi_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

Fibrations between mapping spaces

We study the behaviour of fibre maps under exponentiation, i.e. given a fibration p:E→B we ask for which spaces X is the induced map between mapping spaces p⁎:EX→BX also a fibration. If X is a locally compact space, the positive answer follows easily by the exponential law so in this paper we consider more general spaces and show that the preservation of fibrations is related to the local homotopy properties of the space X. For example, if p is a Dold fibration and X admits a deformation retraction on a compactly generated space, then the induced map p⁎ is also a Dold fibration. Similar results hold for Hurewicz fibrations with unique path-lifting and for covering spaces, and can be furthermore extended to spaces that admit some numerable cover, whose elements preserve fibration property.

Well-rounded equivariant deformation retracts of Teichmüller spaces

In this paper, we construct spines , i.e., \mathrm {Mod}_g -equivariant deformation retracts, of the Teichmüller space \mathcal T_g of compact Riemann surfaces of genus g . Specifically, we define a \mathrm {Mod}_g -stable subspace S of positive codimension and construct an intrinsic \mathrm {Mod}_g -equivariant deformation retraction from mathcal T_g to S . As an essential part of the proof, we construct a canonical \mathrm {Mod}_g -deformation retraction of the Teichmüller space \mathcal T_g to its thick part \mathcal T_g(\varepsilon) when \varepsilon is sufficiently small. These equivariant deformation retracts of \mathcal T_g give cocompact models of the universal space \underline{E}\mathrm {Mod}_g for proper actions of the mapping class group \mathrm {Mod}_g . These deformation retractions of \mathcal T_g are motivated by the well-rounded deformation retraction of the space of lattices in \mathbb R^n . We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.

Topology of moduli spaces of free group representations in real reductive groups

Abstract Let G be a real reductive algebraic group with maximal compact subgroup K, and let Fr be a rank r free group. We show that the space of closed orbits in Hom(Fr ,G)/G admits a strong deformation retraction to the orbit space Hom(Fr ,K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G, such as Sp(4,ℝ) and U(2,2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

Functional and computed tomographic evolution and survival of restrictive allograft syndrome after lung transplantation

Restrictive allograft syndrome (RAS) has recently been defined as a novel phenotype of chronic lung allograft dysfunction (CLAD) after lung transplantation. The goal was to describe computed tomographic (CT) changes of RAS patients and to correlate this with spirometry and survival. All 24 established RAS patients at our center were retrospectively included. CT scans from pre-CLAD, CLAD, post-CLAD and late-CLAD subjects were systematically evaluated by a blinded observer using a semi-quantitative scoring system. Changes in CT patterns were correlated with spirometry and survival. The most prominent CT features at diagnosis of CLAD as compared with pre-CLAD were appearance of central (p = 0.020) and peripheral ground glass opacities (p = 0.052), as well as septal and non-septal lines (p = 0.020). Survival after diagnosis of CLAD was only associated with the absolute value of forced vital capacity (FVC) at diagnosis (R = 0.46 and p = 0.021), and not with any CT alterations. Evolution of CT abnormalities after diagnosis of CLAD included significant increases in (traction) bronchiectasis (p < 0.0001), central (p = 0.051) and peripheral (p = 0.0002) consolidation, architectural deformation (p = 0.0002), volume loss (p = 0.0004) and hilus retraction (p = 0.0036). The absolute FVC decrease post-CLAD diagnosis correlated with CT alterations. In the early stages of RAS, central and peripheral ground glass opacities are the most prominent feature on CT, whereas, in later stages, bronchiectasis, traction, central and peripheral consolidation, architectural deformation, volume loss and hilus retraction are more pronounced. CT changes, however, could not predict survival, whereas FVC at diagnosis of CLAD seems to be the best predictor of survival.

Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μź(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.

Fuzzy Retraction of Fuzzy Space Time

Our aim in the present article is to introduce and study a new type of metric, namely fuzzy space time metric. The geodesics of fuzzy space time metric will be obtained from the view point of Lagrangian equations. Types of the fuzzy retraction of fuzzy space time metric are presented. Types of the fuzzy folding of fuzzy space time metric are defined and discussed. The deformation retraction is also discussed. Some applications concerning these relations are presented. Keywords: Fuzzy Space Time Metric, Fuzzy Retraction, Fuzzy Deformation Retraction, Fuzzy Geodesics, Fuzzy Folding. 2000 Mathematics Subject Classification: 53A35, 51H05, 58C05.

Computational homotopy of finite regular CW-spaces

We describe a computational approach to the basic homotopy theory of finite regular CW-spaces, paying particular attention to spaces arising as a union of closures of $$n$$ -cells with isomorphic face posets. Deformation retraction methods for computing fundamental groups, integral homology and persistent homology are given in this context. The approach is not new but our account contains some new features: (i) certain computational advantages of a permutahedral face poset are identified and utilized; (ii) the notion of lattice complex is introduced as a data type for implementing a general class of regular CW-spaces (including certain pure cubical and pure permutahedral subspaces of flat manifolds such as $$\mathbb{R }^n$$ ); (iii) zig-zag homotopy retractions are introduced as an initial procedure for reducing the number of cells of low-dimensional lattice spaces; more standard discrete vector field techniques are applied for further cellular reduction; (iv) a persistent homology approach to feature recognition in low-dimensional digital images is illustrated; (v) fundamental groups are computed; (vi) algorithms are implemented in the gap system for computational algebra, allowing for their output to benefit from the system’s vast library of efficient algebraic procedures.

An explicit retraction of symplectic flag manifolds onto complex flag manifolds

We construct an explicit deformation retraction of the manifold of symplectic flags onto the manifold of complex flags. The main tool is the polar decomposition of symplectic matrices. We also give a new definition of symplectic Stiefel manifold and prove that it has the same homotopy type as the complex Stiefel manifold.

Foldings and Deformation Retractions of Hypercylinder

Our aim in the present work is to introduce and study new types of retractions of hypercylinder. Types of the deformation retracts of hypercylinder are presented. The relations between the folding and the deformation retract of hypercylinder are deduced. Types of minimal retractions of hypercylinder are introduced. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding are obtained. New types of homotopy maps are deduced. Theorems governing this connection are achieved.