Equivariant geometric bordism, representation, labelled graph

TL;DR

This paper investigates which $G_k$-representation polynomials can be realized as fixed point data of $G_k$-manifolds, using GKM theory and representation theory, and classifies certain 4-manifolds with $G_3$-actions.

Contribution

It introduces a new approach combining GKM theory and representation theory to characterize fixed point data and classifies 4-dimensional manifolds with specific group actions.

Findings

Characterization of fixed point data via $G_k$-labelled graphs.

Complete classification of 4-manifolds with $G_3$-actions fixing finite sets.

Connection between fixed point data and $G_k$-representation polynomials.

Abstract

This paper focuses on the following problem: {\em what $G_k$-representation polynomials in Conner--Floyd $G_k$-representation algebra arise as fixed point data of $G_k$-manifolds?} where $G_k=(\mathbb{Z}_2)^k$. Using the idea of the GKM theory, we construct a $G_k$-labelled graph from a smooth closed manifold with an effective $G_k$-action fixing a finite set. Then we give an answer to above mentioned problem through two approaches: $G_k$-labelled graphs and $G_k$-representation theory. As an application, we give a complete classification of all 4-dimensional smooth closed manifolds with an effective $G_3$-action fixing a finite set up to equivariant unoriented bordism.