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Equivariant geometric bordism, representation, labelled graph

Hao Li, Zhi Lü, Qifan Shen

TL;DR

This work resolves which polynomials in the Conner–Floyd $G_k$-representation algebra arise as fixed-point data of $G_k$-manifolds by developing two complementary frameworks: $G_k$-labelled graphs (inspired by GKM theory) and $G_k$-representation theory. It proves a localization criterion linking fixed-point data to integrality in $H^*(BG_k;\mathbb Z_2)$ and provides a combinatorial description of $\mathcal S_*(G_k)$ via abstract graphs, then a representation-based characterization that complements the graph approach. The paper then applies these tools to the 4-dimensional case with $k=3$, establishing $\dim_{\mathbb Z_2}\mathcal S_4(G_3)=32$ and $\dim_{\mathbb Z_2}\mathcal Z_4(G_3)=32$, and giving explicit generators both algebraically (three essential families $\Lambda_1,\Lambda_2,\Lambda_3$) and geometrically (realized by manifolds such as ${\mathbb R}P^2\times{\mathbb R}P^2$, ${\mathbb R}P^4$, and a real small-cover). These results yield a complete equivariant unoriented bordism classification in dimension four for the $G_3$-action and illustrate the deep link between fixed-point data, graph combinatorics, and $G_k$-representations. The methods pave the way for systematic fixed-point data analysis in higher-dimensional equivariant bordism via the dual graph- and representation-theoretic viewpoints.

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.

Equivariant geometric bordism, representation, labelled graph

TL;DR

This work resolves which polynomials in the Conner–Floyd -representation algebra arise as fixed-point data of -manifolds by developing two complementary frameworks: -labelled graphs (inspired by GKM theory) and -representation theory. It proves a localization criterion linking fixed-point data to integrality in and provides a combinatorial description of via abstract graphs, then a representation-based characterization that complements the graph approach. The paper then applies these tools to the 4-dimensional case with , establishing and , and giving explicit generators both algebraically (three essential families ) and geometrically (realized by manifolds such as , , and a real small-cover). These results yield a complete equivariant unoriented bordism classification in dimension four for the -action and illustrate the deep link between fixed-point data, graph combinatorics, and -representations. The methods pave the way for systematic fixed-point data analysis in higher-dimensional equivariant bordism via the dual graph- and representation-theoretic viewpoints.

Abstract

This paper focuses on the following problem: {\em what -representation polynomials in Conner--Floyd -representation algebra arise as fixed point data of -manifolds?} where . Using the idea of the GKM theory, we construct a -labelled graph from a smooth closed manifold with an effective -action fixing a finite set. Then we give an answer to above mentioned problem through two approaches: -labelled graphs and -representation theory. As an application, we give a complete classification of all 4-dimensional smooth closed manifolds with an effective -action fixing a finite set up to equivariant unoriented bordism.
Paper Structure (16 sections, 21 theorems, 85 equations)

This paper contains 16 sections, 21 theorems, 85 equations.

Table of Contents

  1. introduction
  2. $G_k$-labelled graphs and tom Dieck--Kosniowski--Stong localization Theorem
  3. Construction of $G_k$-labelled graphs from $G_k$-manifolds
  4. Abstract $G_k$-labelled graphs
  5. tom Dieck--Kosniowski--Stong localization theorem
  6. Pure combinatorial description of $\mathcal{S}_*(G_k)$
  7. Characterization of all polynomials of $\mathcal{S}_*(G_k)$ in terms of $G_k$-labelled graphs
  8. Characterization of all polynomials of $\mathcal{S}_*(G_k)$ in terms of $G_k$-representations
  9. An application to $\mathcal{Z}_{4}(G_3)$
  10. Essential generators of $\mathcal{S}_n(G_k)$
  11. Types of faithful $G_k$-representations in $\mathcal{R}_{k+1}(G_k)$
  12. Determination of $\mathcal{S}_4(G_3)$
  13. Geometric generators of $\mathcal{Z}_4(G_3)$
  14. Geometric generator corresponding to $\Lambda_1$
  15. Geometric generator corresponding to $\Lambda_2$
  16. ...and 1 more sections

Key Result

Theorem 2.3

In the extreme case when $n=k$, as mentioned before, a different characterization in terms of the vanishing of a differential on the duals of all polynomials in $\mathcal{S}_k(G_k)$ was given in LT. Darby showed in Darby1Darby2 that a similar characterization for the tangential $T^k$-representations

Theorems & Definitions (40)

  • Remark 1
  • Remark 3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: tom Dieck--Kosniowski--Stong
  • Corollary 2.5
  • Lemma 2.6
  • Corollary 2.7
  • Definition 3.1
  • ...and 30 more