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On the equivariant triangulation of some small covers

Raju Kumar Gupta, Soumen Sarkar

TL;DR

The paper addresses the problem of determining minimal ${\mathbb{Z}}_2^n$-equivariant triangulations of small covers and how these triangulations relate to the orbit space. It develops a general framework showing that such triangulations induce triangulations on orbit spaces, and then leverages this to construct explicit minimal equivariant triangulations: a unique ${\mathbb{Z}}_2^3$-equivariant triangulation of ${\mathbb{RP}}^3$ with $11$ vertices, and a ${\mathbb{Z}}_2^3$-equivariant triangulation of ${\mathbb{RP}}^3 \# {\mathbb{RP}}^3$ with $17$ vertices that yields a new minimal $g$-vector $(12,35)$. The authors also present a method for building equivariant triangulations of connected sums and apply it to obtain further bounds, while proving a nonexistence result for ${\mathbb{RP}}^4$ under ${\mathbb{Z}}_2^4$ action at up to $17$ vertices. Overall, the work advances the understanding of how symmetry constraints shape minimal triangulations of real projective spaces and their connected sums, with precise vertex-count and combinatorial constraints tied to the group action.

Abstract

In this paper, we study certain properties of $\mathbb{Z}_2^n$-equivariant triangulations of small covers. We show that any $\mathbb{Z}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$, which contains $11$ vertices and prove that this is the unique $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ with $17$ vertices, which is the best known yet. This triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a $\ZZ_2^4$-equivariant triangulation of $\mathbb{RP}^4$ requires at least $18$ vertices.

On the equivariant triangulation of some small covers

TL;DR

The paper addresses the problem of determining minimal -equivariant triangulations of small covers and how these triangulations relate to the orbit space. It develops a general framework showing that such triangulations induce triangulations on orbit spaces, and then leverages this to construct explicit minimal equivariant triangulations: a unique -equivariant triangulation of with vertices, and a -equivariant triangulation of with vertices that yields a new minimal -vector . The authors also present a method for building equivariant triangulations of connected sums and apply it to obtain further bounds, while proving a nonexistence result for under action at up to vertices. Overall, the work advances the understanding of how symmetry constraints shape minimal triangulations of real projective spaces and their connected sums, with precise vertex-count and combinatorial constraints tied to the group action.

Abstract

In this paper, we study certain properties of -equivariant triangulations of small covers. We show that any -equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal -equivariant triangulation of , which contains vertices and prove that this is the unique -equivariant triangulation of with vertices. For a finite group , we give a method for constructing some -equivariant triangulations of connected sums of manifolds from their respective -equivariant triangulations. In particular, we construct a -equivariant triangulation of with vertices, which is the best known yet. This triangulation of provides another minimal -vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a -equivariant triangulation of requires at least vertices.
Paper Structure (6 sections, 26 theorems, 34 equations, 7 figures)

This paper contains 6 sections, 26 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant triangulation of $\mathbb{S}^n$
  4. Equivariant triangulation of a class of small covers
  5. Equivariant Triangulation of Connected Sums
  6. Equivariant Triangulation on $\mathbb{RP}^4$

Key Result

Corollary 2.4

If $X$ is a $G$-equivariant triangulation of an $n$-manifold $M$. Then $G$ is a subgroup of $Aut(X)$.

Figures (7)

  • Figure 1: ${\mathbb{Z}}_2^3$-equivariant triangulations of $D^3$ with $6$ and $7$ vertices.
  • Figure 2: Reflections along a hyperplane.
  • Figure 3: Triangulations of $B_1$ and $B_2$.
  • Figure 4: $\hbox{\upshape st}\,(e,K)$ and $\hbox{\upshape st}\,(f,K)$ respectively.
  • Figure 5: $\hbox{\upshape st}\,(1)$, $\hbox{\upshape st}\,(2)$ and $\hbox{\upshape st}\,(3)$ respectively.
  • ...and 2 more figures

Theorems & Definitions (69)

  • Conjecture 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Corollary 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 59 more