A structure theorem for homology 4-manifolds with $g_2\leq 5$

TL;DR

This paper provides a combinatorial characterization of homology 4-manifolds with g_2 ≤ 5, showing they are triangulated spheres derived from simpler spheres through specific operations, and proves the bound is sharp.

Contribution

It establishes a structural theorem for homology 4-manifolds with g_2 ≤ 5, detailing their construction from triangulated 4-spheres with g_2 ≤ 2.

Findings

Homology 4-manifolds with g_2 ≤ 2 are polytopal spheres.

Homology 4-manifolds with g_2 ≤ 5 are triangulated spheres derived from g_2 ≤ 2 spheres.

The bound g_2 ≤ 5 is optimal; it cannot be extended to g_2 = 6.

Abstract

Numerous structural findings of homology manifolds have been derived in various ways in relation to $g_2$-values. The homology $4$-manifolds with $g_2\leq 5$ are characterized combinatorially in this article. It is well-known that all homology $4$-manifolds for $g_2\leq 2$ are polytopal spheres. We demonstrate that homology $4$-manifolds with $g_2\leq 5$ are triangulated spheres and are derived from triangulated 4-spheres with $g_2\leq 2$ by a series of connected sum, bistellar 1- and 2-moves, edge contraction, edge expansion, and edge flipping operations. We establish that the above inequality is optimally attainable, i.e., it cannot be extended to $g_2 = 6$.