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On a construction of some homology $d$-manifolds

Biplab Basak, Sourav Sarkar

TL;DR

This work addresses the classification of homology $d$-manifolds with the $g_2$-vector value $g_2=3$ by developing a combinatorial framework based on normal pseudomanifolds, rigidity theory, and a suite of local moves. It proves that such manifolds are necessarily triangulated spheres and provides explicit constructions describing their forms, including joins of lower-$g_2$ spheres, retriangulations, and connected sums from spheres with $g_2 eq3$. The authors also establish a structural result for prime normal $d$-pseudomanifolds with $g_2=3$, detailing how these manifolds can be built from simpler pieces via a sequence of combinatorial operations. Overall, the paper advances a g-vector–driven understanding of the topology of homology manifolds and offers concrete tools for constructing and recognizing these objects in higher dimensions.

Abstract

The $g$-vector of a simplicial complex contains a lot of information about the combinatorial and topological structure of that complex. Several classification results regarding the structure of normal pseudomanifolds and homology manifolds have been established concerning the value of $g_2$. It is known that when $g_2=0$, all normal pseudomanifolds of dimensions at least three are stacked spheres. In the cases of $g_2=1$ and $2$, all homology manifolds are polytopal spheres and can be obtained through retriangulation or join operations from the previous ones. In this article, we provide a combinatorial characterization of the homology $d$-manifolds, where $d\geq 3$ and $g_2=3$. These are spheres and can be obtained through operations such as joins, some retriangulations, and connected sums from spheres with $g_2\leq 2$. Furthermore, we have presented a structural result on prime normal $d$-pseudomanifolds with $g_2=3$.

On a construction of some homology $d$-manifolds

TL;DR

This work addresses the classification of homology -manifolds with the -vector value by developing a combinatorial framework based on normal pseudomanifolds, rigidity theory, and a suite of local moves. It proves that such manifolds are necessarily triangulated spheres and provides explicit constructions describing their forms, including joins of lower- spheres, retriangulations, and connected sums from spheres with . The authors also establish a structural result for prime normal -pseudomanifolds with , detailing how these manifolds can be built from simpler pieces via a sequence of combinatorial operations. Overall, the paper advances a g-vector–driven understanding of the topology of homology manifolds and offers concrete tools for constructing and recognizing these objects in higher dimensions.

Abstract

The -vector of a simplicial complex contains a lot of information about the combinatorial and topological structure of that complex. Several classification results regarding the structure of normal pseudomanifolds and homology manifolds have been established concerning the value of . It is known that when , all normal pseudomanifolds of dimensions at least three are stacked spheres. In the cases of and , all homology manifolds are polytopal spheres and can be obtained through retriangulation or join operations from the previous ones. In this article, we provide a combinatorial characterization of the homology -manifolds, where and . These are spheres and can be obtained through operations such as joins, some retriangulations, and connected sums from spheres with . Furthermore, we have presented a structural result on prime normal -pseudomanifolds with .
Paper Structure (6 sections, 34 theorems, 12 equations, 1 figure)

This paper contains 6 sections, 34 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal pseudomanifolds
  4. Concept of Rigidity
  5. Structural tools to be used
  6. Proof of the Main Theorem

Key Result

Proposition 1.1

NevoNovinsky Let $d\geq 3$, and let $\Delta$ be a prime homology $d$-sphere with $g_2(\Delta)=1$. Then $\Delta$ is combinatorially isomorphic to either the join of the boundary complex of two simplices, where each simplex has a dimension of at least $2$ and their dimensions add up to $d+1$, or the j

Figures (1)

  • Figure 1: The antipodal vertices $x$ and $y$ in $lk_{\Delta} u$.

Theorems & Definitions (62)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5: BGWalkup
  • Theorem 1.6
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Lemma 2.4
  • ...and 52 more