arXiv research

This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.

Contact group retracts to unitary subgroup.

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichmüller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$…

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

The paper creates a deformation retraction for homeomorphisms of the projective plane.

Embeds complex into higher-dimensional pseudomanifold.

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.

Simplicial sets deformation retract onto transverse simplices.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.

Global homotopies upgrade classical map in differential geometry.

Spaces of circle embeddings in curved surfaces indexed by trees.

Schmutz Schaller and Thurston's approaches are dual.

Control data constructed for smooth weak deformation retraction of stratified spaces.

Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.

Let f:Σ_1 --> Σ_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ_1\times Σ_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through…

Constructs Teichmüller curve to study Thurston spine structure.

We construct a one-dimensional deformation retract of the unordered k-point configuration space of a star S. This retract suggests an explicit set of free generators Beta_k for the corresponding braid group of the star B_k and shows that the natural map from B_k-1 to B_k sends Beta_k-1 to Beta_k injectively.

Let $Γ=\langle a,b | a b^{p} a^{-1} = b^{q}\rangle$ be a Baumslag--Solitar group and $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, there is a strong deformation retraction retraction of $Hom(Γ,G)$ onto $…

Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γgenerated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to …

We present a (possibly) new sphere eversion based on the contractibility* of a certain subset of the space of immersions of the circle in the plane. (*: by strong deformation retraction)

Selects points from Jordan domains on Riemannian surfaces.

In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of the 3-sphere S^3 deformation retracts to the subspace of Hopf fibrations, and so has the homotopy type of a pair of disjoint two-spheres. Since that time, no generalization of this result to higher dimensions has been found, a…

This study examines the topology of singularities in optimal semicouplings between unequal spaces.

Constructs a moment map flow for isotropic maps on surfaces.

Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.

Study the boundary of a space related to Outer space.

We prove that the set of symplectic lattices in the Siegel space $\mathfrak{h}_g$ whose systoles generate a subspace of dimension at least 3 in $\mathbb{R}^{2g}$ does not contain any $\mathrm{Sp}(2g,\mathbb{Z})$-equivariant deformation retract of $\mathfrak{h}_g$.

In this short note, we investigate some features of the space $\Inject{d}{m}$ of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the set of orthonormal systems of $d$ vectors in $\bbR^m$. Finally, we…

Study on higher-dimensional black holes, focusing on retractions and scalar quasibound states.

The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.

The paper shows conditions under which certain 4-manifolds have no smooth spines.

Researchers describe character varieties for Hopf links, proving geometric properties.

The paper proves parabolic gap theorems for Yang-Mills energy.

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…

New contractible complex shows virtual cohomological dimension of RAAGs.

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem…

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…

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

Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate the fundamental group of the torus is, for n > 4, not contractible. In particular, Y is not a…

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent …

A new retraction on Stiefel manifold with a closed-form inverse.

New result on critical points of Bethe free energy under deformation retracts.

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independen…

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…

Computational method approximates homology groups of compact metric spaces.

We study Poincaré type $L^p$ inequality on a compact semialgebraic subset of $\R^n$ for $p>>1$. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives. Then, we extend the local inequality to a global inequality by employing double complex technique. As a conseq…

New retraction on symplectic Stiefel manifold with closed-form inverse.