Graphs for torus actions on oriented manifolds with isolated fixed points and classification in dimension 6

TL;DR

This paper introduces a graph-based method to classify torus actions on oriented manifolds with isolated fixed points, focusing on 6-dimensional cases and their fixed point data transformations.

Contribution

It develops a classification framework for fixed point data of torus actions on 6-manifolds using multigraphs and operations like connected sums and blow-ups.

Findings

Classified multigraphs for circle actions on 6-manifolds.

Established methods to reduce fixed point data to an empty graph.

Extended classification to 4-dimensional manifolds.

Abstract

Let a torus act on a compact oriented manifold $M$ with isolated fixed points, with an additional mild assumption that its isotropy submanifolds are orientable. We associate a signed labeled multigraph encoding the fixed point data (weights and signs at fixed points and isotropy submanifolds) of the manifold. We study operations on $M$ and its multigraph, (self) connected sum and blow up, etc. When the circle group acts on a 6-dimensional $M$, we classify such a multigraph by proving that we can convert it into the empty graph by successively applying two types of operations. In particular, this classifies the fixed point data of any such manifold. We prove this by showing that for any such manifold, we can successively take equivariant connected sums at fixed points with itself, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$ of the Hirzebruch surfaces (and these with opposite orientations) to a fixed point free action on a compact oriented 6-manifold. We also classify a multigraph for a torus action on a 4-dimensional $M$.