Related papers: Topological symmetry groups and mapping class grou…
This article is covered by the article arxiv.1012.0925 We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n…
Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (resp. a symplectic form) for which u is orthogonal (resp.…
We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be…
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
We propose an algorithm to estimate the topology of an embedded metric graph from a well-sampled finite subset of the underlying graph.
We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover…
We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…
We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle $\mathrm{C}_n^{(\lambda)}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n\geqslant 3$ and…
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a…
In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.
We solve the isomorphism problem for the whole class of Lins-Mandel gems (graphs encoded manifolds). We also present certain homeomorphisms of branched cyclic coverings of two-bridge hyperbolic links. As a consequence, we prove that, in in…
Given two tuples of subspaces, can you tell whether the tuples are isomorphic? We develop theory and algorithms to address this fundamental question. We focus on isomorphisms in which the ambient vector space is acted on by either a unitary…
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…
Three-dimensional catalogues of objects at cosmological distances can potentially yield candidate topologically lensed pairs of sets of objects, which would be a sign of the global topology of the Universe. In the spherical case, a…
In this paper, we introduce the notion of a dual topological graph of a given topological graph, and show that it defines a C*-algebra isomorphic to the C*-algebra of the given one. Repeating to take a dual, and taking a projective limit,…
We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle…
Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding…
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for…