[Paper Review] A note on the Nielsen realization problem for K3 surfaces

David Baraglia, Hokuto Konno|arXiv (Cornell University)|Aug 11, 2019

Geometric and Algebraic Topology13 references4 citations

TL;DR

This paper resolves key aspects of the Nielsen realization problem for K3 surfaces using Seiberg-Witten theory and the global Torelli theorem. It proves that while the mapping class group of a K3 surface admits a section into its diffeomorphism group, certain finite subgroups (of order 2) cannot be realized smoothly but can be realized continuously, and that the fundamental group of the diffeomorphism group maps non-surjectively to that of the homeomorphism group.

ABSTRACT

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map $π_{0}(Diff(K3)) o Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which there is no splitting of the map $Diff(K3) o π_{0}(Diff(K3))$, but there is a splitting of $Homeo(K3) o π_{0}(Homeo(K3))$ over the image of $G$ in $π_{0}(Homeo(K3))$, which is non-trivial. The third is that the map $π_{1}(Diff(K3)) o π_{1}(Homeo(K3))$ is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for $K3$ surfaces.

Motivation & Objective

• To investigate the smooth and continuous Nielsen realization problems for K3 surfaces.
• To determine whether finite subgroups of the mapping class group of a K3 surface can be realized as subgroups of the diffeomorphism or homeomorphism group.
• To analyze the homotopy groups of the diffeomorphism and homeomorphism groups of a K3 surface.
• To clarify the obstruction to smoothability of continuous families of K3 surfaces.

Proposed method

• Uses the global Torelli theorem for K3 surfaces to construct a section of the natural map from the diffeomorphism group to the automorphism group of the second cohomology lattice.
• Applies the adjunction inequality from Seiberg-Witten theory to obstruct smooth lifts of certain finite group actions.
• Constructs explicit examples of continuous families of K3 surfaces over a 2-torus that are not smoothable, using obstruction theory on the 2-skeleton of the torus.
• Analyzes the difference obstruction in cohomology groups $ H^2(T^2; ho_1) $ to detect non-isomorphism between smoothable and non-smoothable families.
• Relies on the homotopy quotient $ Q = ext{Homeo}(X) imes_{ ext{Diff}(X)} E ext{Diff}(X) $ to relate obstructions in $ ext{Homeo}(X) $ and $ ext{Diff}(X) $.
• Uses obstruction theory on the 2-skeleton of $ T^2 $ to show that the obstruction class in $ H^2(T^2; ho_1) $ maps non-trivially to $ H^2(T^2; ho_1(Q)) $, implying non-surjectivity of $ \pi_1(\text{Diff}(X)) \to \pi_1(\text{Homeo}(X)) $.

Experimental results

Research questions

• RQ1 Can every finite subgroup of the mapping class group of a K3 surface be realized as a subgroup of the diffeomorphism group?
• RQ2 Is there a finite subgroup of the mapping class group that lifts to the homeomorphism group but not to the diffeomorphism group?
• RQ3 Is the induced map $ \pi_1(\text{Diff}(X)) \to \pi_1(\text{Homeo}(X)) $ surjective?
• RQ4 What is the role of Seiberg-Witten theory in obstructing smooth realizations of group actions on K3 surfaces?
• RQ5 How do the homotopy groups of the diffeomorphism and homeomorphism groups of a K3 surface compare?

Key findings

• There exists a section $ s: \Gamma \to \text{Mod}(X) $, showing that the natural map $ \pi_0(\text{Diff}(X)) \to \text{Aut}(H^2(X;\mathbb{Z})) $ has a section over its image.
• There exists a subgroup $ G \subset \pi_0(\text{Diff}(X)) $ of order 2 that does not lift to a subgroup of order 2 in $ \text{Diff}(X) $, but its image in $ \pi_0(\text{Homeo}(X)) $ does lift in $ \text{Homeo}(X) $.
• The map $ \pi_1(\text{Diff}(X)) \to \pi_1(\text{Homeo}(X)) $ is not surjective, implying $ \pi_1(\text{Homeo}(X)) $ is non-trivial.
• The obstruction to smoothability of a continuous family of K3 surfaces over $ T^2 $ lies in $ H^2(T^2; \pi_1(\text{Homeo}(X))) $, and this obstruction class maps non-trivially to $ H^2(T^2; \pi_1(Q)) $, where $ Q $ is the homotopy fiber of $ B\text{Diff}(X) \to B\text{Homeo}(X) $.
• The non-smoothability of a specific continuous family of K3 surfaces demonstrates that the inclusion $ \text{Diff}(X) \to \text{Homeo}(X) $ induces a non-surjective map on $ \pi_1 $.
• The results are derived from a combination of the global Torelli theorem and Seiberg-Witten theory, particularly the adjunction inequality.