[Paper Review] Diffeomorphism of simply connected algebraic surfaces

Fabrizio Catanese, Bronisław Wajnryb|ArXiv.org|May 14, 2004

Geometric and Algebraic Topology23 references4 citations

TL;DR

This paper constructs explicit families of simply connected minimal algebraic surfaces of general type that are diffeomorphic yet not deformation equivalent, providing a counterexample to a weakened form of the Friedman-Morgan conjecture (DEF = DIFF) in the simply connected case. Using simple bidouble covers of P¹×P¹ with specific numerical invariants, the authors prove diffeomorphism via symplectic Lefschetz fibrations and monodromy factorizations in the mapping class group, showing that distinct deformation types can share the same smooth and symplectic structure.

ABSTRACT

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where (by M. Freedman's theorem) the topological type is completely determined by the numerical invariants of the surface. We hope that the methods of proof may turn out to be quite useful to show diffeomorphism and indeed symplectic equivalence for many important classes of algebraic surfaces and symplectic 4-manifolds.

Motivation & Objective

• To disprove a weakened form of the Friedman-Morgan conjecture (DEF = DIFF) for simply connected minimal algebraic surfaces of general type.
• To construct explicit families of such surfaces that are diffeomorphic but not deformation equivalent, thereby demonstrating that smooth structure does not determine deformation type.
• To establish a method for proving diffeomorphism of algebraic surfaces using symplectic Lefschetz fibrations and monodromy factorizations in the mapping class group.
• To explore the possibility of symplectomorphism for these surfaces under the canonical symplectic structure, linking the problem to braid group factorizations.

Proposed method

• Constructing surfaces as simple bidouble covers of P¹×P¹ with specified ramification types (2a,2b), (2c,2b) and their perturbations (2a+2k,2b), (2c−2k,2b).
• Using the holomorphic fibration over P¹ to realize the surfaces as symplectic Lefschetz fibrations via symplectic perturbation of the fibration map.
• Reducing the diffeomorphism problem to comparing two factorizations of the identity in the mapping class group of the fiber surface.
• Applying a lemma from Auroux (2002) to show that the glueing diffeomorphism ψ is a product of Dehn twists, thereby proving diffeomorphism.
• Analyzing the monodromy action on curves (α₁, β₁, γ₁, δ₁, σ) through successive transformations to verify the factorization condition.
• Investigating whether symplectomorphism holds by extending the monodromy argument to the braid group, particularly for the involution ι on P¹ that lifts to ψ.

Experimental results

Research questions

• RQ1 Can there exist simply connected minimal algebraic surfaces of general type that are diffeomorphic but not deformation equivalent?
• RQ2 Does the canonical class remain preserved under diffeomorphisms in the simply connected case, and can such diffeomorphisms be constructed explicitly?
• RQ3 Can symplectic Lefschetz fibrations and monodromy factorizations in the mapping class group be used to prove diffeomorphism of algebraic surfaces?
• RQ4 Is the diffeomorphism type of a surface sufficient to determine its deformation type in the simply connected case of general type surfaces?
• RQ5 Can the monodromy factorization argument be extended to prove symplectomorphism by lifting to the braid group and analyzing the involution on P¹?

Key findings

• For each positive integer h, the paper constructs h pairwise non-deformation-equivalent, simply connected minimal algebraic surfaces of general type that are all diffeomorphic to each other.
• Theorem 2.5 establishes that bidouble covers of P¹×P¹ with types ((2a,2b),(2c,2b)) and ((2a+2k,2b),(2c−2k,2b)) are not deformation equivalent under the given numerical constraints on a,b,c,k.
• Theorem 3.1 proves that surfaces of types ((2a,2b),(2c,2b)) and ((2a+2,2b),(2c−2,2b)) are diffeomorphic when a,b,c−1≥2.
• The diffeomorphism is established by showing that the glueing diffeomorphism ψ in the mapping class group is a product of Dehn twists, using Auroux’s lemma.
• The surfaces are realized as symplectic Lefschetz fibrations over P¹, and their diffeomorphism type is encoded in equivalent monodromy factorizations.
• The authors conjecture that these surfaces may be symplectomorphic under the canonical symplectic structure, but this remains open due to technical difficulties in lifting the monodromy argument to the braid group.