Related papers: Embedding simply connected 2-complexes in 3-space

We show that for $d\geq 2$ every finite $d$-dimensional simplicial complex is a deformation retract of a $(2d-1)$-dimensional pseudomanifold with boundary. Moreover, it embeds as a retract in a closed $(2d-1)$-dimensional pseudomanifold.

Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of $K_5$ and of $K_{3,3}$. Here we study the minimal (under subdigraph containment) strongly-connected digraphs that are…

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus $k$ or for embeddings into nonorientable surface…

For a fixed radius $r$ and a point $o$ in the curve complex of a surface, we define the sphere of radius $r$ to be the induced subgraph on the set of vertices of distance $r$ from $o$. We show that these spheres are almost simply connected…

This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]-[CM5] we describe the case where the surfaces are topologically disks on…

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) $2$-complexes. We show, as an extension of an earlier work, that computing first homology of $2$-complexes is…

Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a…

In this paper we give a proof of Lichnerowicz Conjecture for compact simply connected manifolds which is intrinsic in the sense that it avoids the {\it Nice Embeddings} into eigen spaces of the Laplacian. Even if one wants to use these…

We extend Matveev's theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible 3-orbifolds, and, with certain restrictions and…

We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a…

We classify simple parametrisations of complex curve singularities. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal…

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give…

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show…

We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to…

We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. We then apply the procedure…

Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He…

It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised…

We say that a $2$-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to the $2$-dimensional Euclidean space, then we obtain a $1$-dimensional complex which is homeomorphic to…

Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such…

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space…