Lecture Notes: Exotic Phenomena in Dimension 4

Introduction

• Speaker: Lisa Piccirillo from MIT and UT Austin
• Notable achievements: Solved the Conway knot problem, awarded multiple prestigious fellowships and prizes.
• Topic: Exotic phenomena in 4-dimensions, focusing on manifolds.

Basics of Topology and Manifolds

Types of Manifolds

• Topological Manifolds: A nice topological space with a local atlas to ( \mathbb{R}^n ).
• Smooth Manifolds: Require transition functions to be ( C^\infty ).
• PL Manifolds: Not discussed as they coincide with smooth manifolds in dimension 4.

Convenience Assumptions

• All manifolds considered are oriented, compact, and without boundary unless stated otherwise.

Classification of Manifolds

Low Dimensions (1, 2, 3)

• Classification theorems exist.
• In dimension 3, smooth and topological categories coincide (Moise's theorem).

High Dimensions (≥5)

• Surgery Theory: Developed by Browder, Novikov, Sullivan, and Wall.
  • Reduces classification to algebraic topology.
• Misconception: High-dimensional manifold classification is complete; it's not.
• Example: Smooth Poincaré conjecture is open in dimension 126.
• Difference between smooth and topological categories (Milner's work).

Dimension 4

Topological Category

• Friedman's work (1982): Surgery theory works topologically for 'good' fundamental groups.
• Classification of simply connected topological 4-manifolds via intersection forms.

Smooth Category

• No classification theorems or conjectures.
• A smooth 4-manifold ( x ) is exotic if there exists another smooth 4-manifold ( x' ) which is homeomorphic but not diffeomorphic.
• Donaldson's work proves exotic 4-manifolds exist.

Key Question

• Which smooth 4-manifolds are exotic?
  • Particularly interested in those with trivial fundamental group and small ( b_2 ).

Techniques and Developments

Classical Methods

1. Building Exotic Manifolds:

   • Construct candidates.
   • Show they are homeomorphic (Friedman's intersection form criteria).
   • Show they are not diffeomorphic (gauge theory).

2. Sources of 4-manifolds:

   • Basic examples: 4-sphere, complex projective space, etc.
   • Products and bundles: ( S^2 \times S^2 ), 3-manifold × ( S^1 ).
   • Complex surfaces and symplectic manifolds.
   • Cut and paste operations.
   • Handle constructions (not classically used).

3. Distinguishing Manifolds:

   • Primarily through gauge theory, which counts solutions to a PDE.
   • Limitations: Not computable explicitly in general, doesn't work with ( b_2 = 0 ).

Recent Developments

• 2005-Recent: Rational blowdown technique, arms race to find smaller ( b_2 ) exotica.
• Current Work:
  • 2021: Slice knot approach for distinguishing manifolds (potential for no ( b_2 )).
  • New constructions using explicit handle structures.
  • ( b_2 = 4 ) exotic manifolds identified with certain properties.
  • Recent results providing new techniques and smallest exotica with ( b_2 = 1 ), ( \pi_1 = \mathbb{Z}/2 ).
  • Combinatorial methods for exotic manifolds with boundary through skein lasagna module.

Conclusion

• Exploration of structural origins of exotica and potential quantification methods.
• Discussion of limitations and future directions in the study of smooth 4-manifolds.