Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces

On the embedding dimension of 2-torsion lens spaces

Using the $ku$- and $BP$-theoretic versions of Astey's cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for $e$ greater than or equal to $α(n)$--the number of ones in the dyadic expansion of $n$--, the ($2n+1$)-dimensional $2^e$-torsion lens space cannot be embedded in Euclidean space of dimension $4n-2α(n)+1$. A slightly restricted version of this fact holds for $e<α(n)$. We also give an inductive construction of Euclidean embeddings for $2^e$-torsion lens spaces. Some of our best embeddings are within one dimension of b…

Two Representations of the Fundamential Group and Invariants of Lens Spaces

This article is a continuation of work on construction and calculation various of modifications of invariant based on the use Euclidean metric values attributed to elements of manifold triangulation. We again address the well investigated lens spaces as a standard tool for checking the nontriviality of topological invariants.

Approximating Persistent Homology in Euclidean Space Through Collapses

The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the Čech persistence module. Both are constructed on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistenc…

Symmetric topological complexity of projective and lens spaces

For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and compl…

Distinguishing three-dimensional lens spaces L(7,1) and L(7,2) by means of classical pentagon equation

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7,1) and L(7,2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e.algebraic relation corresponding to a ``2 tetrahedra to 3 tetrahedra'' local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.

三角剖分流形的 Cheeger-Gromov 不变量与单纯复杂度的界

Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds

本文通过 G-着色多面体和相对双曲化构造线性复杂度配边，建立 PL 流形 Wall 与 Cheeger-Gromov ρ-不变量的线性界，并应用于透镜空间 h-配边类计数、高维透镜空间复杂度及结构集非有限生成性。