Decompositions of three-dimensional Alexandrov spaces

TL;DR

This paper extends 3-manifold topology results to Alexandrov 3-spaces, introducing new decompositions, a generalized surgery, and linking these spaces to 4-dimensional Alexandrov spaces.

Contribution

It generalizes prime decomposition and Dehn surgery concepts to Alexandrov 3-spaces, unifying manifold and non-manifold cases.

Findings

Established a prime decomposition theorem for Alexandrov 3-spaces.

Proved that any closed Alexandrov 3-space can be obtained via generalized Dehn surgery.

Showed every closed Alexandrov 3-space is homeomorphic to the boundary of a 4-dimensional Alexandrov space.

Abstract

We extend basic results in $3$-manifold topology to general three-dimensional Alexandrov spaces (or Alexandrov $3$-spaces for short), providing a unified framework for manifold and non-manifold spaces. We generalize the connected sum to non-manifold $3$-spaces and prove a prime decomposition theorem, exhibit an infinite family of closed, prime non-manifold $3$-spaces which are not irreducible, and establish a conjecture of Mitsuishi and Yamaguchi on the structure of closed, simply-connected Alexandrov $3$-spaces with non-negative curvature. Additionally, we define a notion of generalized Dehn surgery for Alexandrov $3$-spaces and show that any closed Alexandrov $3$-space may be obtained by performing generalized Dehn surgery on a link in $S^3$ or the non-trivial $S^2$-bundle over $S^1$. As an application of this result, we show that every closed Alexandrov $3$-space is homeomorphic to the boundary of a $4$-dimensional Alexandrov space.