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Cyclic Nielsen realization for del Pezzo surfaces

Seraphina Eun Bi Lee, Tudur Lewis, Sidhanth Raman

TL;DR

This work advances the understanding of which finite-order mapping classes of the smooth manifolds $M_n=\mathbb{C}P^2\#n\overline{\mathbb{C}P^2}$ can be realized by complex automorphisms, isometries of Einstein metrics, or diffeomorphisms. It builds a Coxeter-theoretic framework via the Weyl group $W_n$ to enumerate irreducible, finite-order classes and to analyze realizability through Carter graphs and fixed-point data. The authors prove an equivalence of metric, complex, Kähler–Einstein, and smooth Nielsen realizations for all irreducible elements with $H_2(M_n,\mathbb{Z})^{\langle f\rangle}\cong \mathbb{Z}$ for $3\le n\le 7$, and provide detailed nonrealizability results for several classes in $M_5$, $M_7$, and $M_8$ using $G$-signature, Edmonds’ theorem, and related tools. They also show that Coxeter elements account for many realizable classes, and classify irreducibles of odd prime order, connecting complex realizability to birational automorphisms and minimal rational $G$-surfaces. The results synthesize 4-manifold topology, algebraic geometry, and Coxeter theory to map the landscape of Nielsen realizations in this rich geometric setting.

Abstract

The cyclic Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order can be represented by a homeomorphism of the same order. In this article, we resolve the smooth, metric, and complex cyclic Nielsen realization problem for certain "irreducible" mapping classes on the family of smooth 4-manifolds underlying del Pezzo surfaces. Both positive and negative examples of realizability are provided in various settings. Our techniques are varied, synthesizing results from reflection group theory and 4-manifold topology.

Cyclic Nielsen realization for del Pezzo surfaces

TL;DR

This work advances the understanding of which finite-order mapping classes of the smooth manifolds can be realized by complex automorphisms, isometries of Einstein metrics, or diffeomorphisms. It builds a Coxeter-theoretic framework via the Weyl group to enumerate irreducible, finite-order classes and to analyze realizability through Carter graphs and fixed-point data. The authors prove an equivalence of metric, complex, Kähler–Einstein, and smooth Nielsen realizations for all irreducible elements with for , and provide detailed nonrealizability results for several classes in , , and using -signature, Edmonds’ theorem, and related tools. They also show that Coxeter elements account for many realizable classes, and classify irreducibles of odd prime order, connecting complex realizability to birational automorphisms and minimal rational -surfaces. The results synthesize 4-manifold topology, algebraic geometry, and Coxeter theory to map the landscape of Nielsen realizations in this rich geometric setting.

Abstract

The cyclic Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order can be represented by a homeomorphism of the same order. In this article, we resolve the smooth, metric, and complex cyclic Nielsen realization problem for certain "irreducible" mapping classes on the family of smooth 4-manifolds underlying del Pezzo surfaces. Both positive and negative examples of realizability are provided in various settings. Our techniques are varied, synthesizing results from reflection group theory and 4-manifold topology.

Paper Structure

This paper contains 26 sections, 44 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Related work
  4. Outline
  5. Acknowledgements
  6. Coxeter theory of $\mathop{\mathrm{Mod}}\nolimits(M_n)$
  7. Topological mapping class groups of del Pezzo manifolds
  8. $\mathop{\mathrm{O}}\nolimits^+(1,n)(\mathbb{Z})$ is a Coxeter group for $2 \leq n \leq 9$.
  9. Weyl group
  10. Irreducibility, parabolic subgroups, and cuspidal conjugacy classes
  11. Cuspidal conjugacy classes of $P_n$
  12. Complex Nielsen realization via enumeration
  13. Irreducibility and minimality
  14. Carter graphs and complex non-realizability
  15. Comparing metric and complex Nielsen realization
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $3 \leq n \leq 7$ and let $f\in \mathop{\mathrm{Mod}}\nolimits^+(M_n)$ be irreducible and of order $m < \infty$. Suppose further that $H_2(M_n, \mathbb{Z})^{\langle f \rangle} \cong \mathbb{Z}$. The following are equivalent:

Figures (1)

  • Figure 1: Coxeter diagrams for $(W(n), S(n))$. The subgraph spanned by the vertices $s_0, \dots, s_{n-1}$ is the Coxeter diagram for the Weyl group $W_n.$

Theorems & Definitions (102)

  • Theorem 1.1: Equivalence of Nielsen Realizations
  • Remark 1.2: On the condition $H_2(M_n, \mathbb{Z})^{\langle f \rangle} \cong \mathbb{Z}$
  • Theorem 1.4: Coxeter Nielsen Realization
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.1
  • Definition 2.2: lee-involutions
  • Theorem 2.3: Wall 1964, wall-indefinite-orthogonal
  • Definition 2.4
  • Definition 2.5
  • ...and 92 more