Will we ever classify simply-connected smooth 4-manifolds?

Abstract

These notes are adapted from two talks given at the 2004 Clay Institute Summer School on Floer homology, gauge theory, and low dimensional topology at the Alfred Renyi Institute. We will quickly review what we do and do not know about the existence and uniqueness of smooth and symplectic structures on closed, simply-connected 4-manifolds. We will then list the techniques used to date and capture the key features common to all these techniques. We finish with some approachable questions that further explore the relationship between these techniques and whose answers may assist in future advances towards a classification scheme.

17 pages, 2004 Clay Institute Summer School on Floer homology, gauge theory, and low dimensional topology. More errata corrected

References in corpus (5)

• The Seiberg-Witten invariants and 4-manifolds with essential tori
• Superconformal invariance and the geography of four-manifolds
• Invariants for Lagrangian tori
• On Park's exotic smooth four-manifolds
• Rational surfaces and symplectic 4-manifolds with one basic class

Cited by in corpus (8)

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• The Partition Function Of Argyres-Douglas Theory On A Four-Manifold
• Constructions of small symplectic 4-manifolds using Luttinger surgery
• On the geography and botany of irreducible nonspin 4-manifolds with abelian fundamental group
• On the existence of Parseval frames for vector bundles
• Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds
• Essential minimal volume of Einstein 4-manifolds