MATRIX 2019 Interactions: Exercises

This page lists the exercises for consideration during the MATRIX 2019 Interactions meeting.

Participants are encouraged to work on the exercises and contribute solutions on the discussion page.

1 Surgery: high-d methods in low-d

1.1 Lecture 1: Normal maps and the surgery obstruction

1. Stability of vector bundles (Ex)
2. Normal maps - (non)-examples (Ex)
3. Immersing n-spheres in 2n-space (Ex)
4. Surgery obstruction, signature (Ex)
5. Surgery obstruction, Arf-invariant (Ex)

1.2 Lecture 2: Foundations of topological 4-manifolds

1. Connected sum of topological manifolds (Ex)
2. Quillen plus construction (Ex)
3. Stability of the E8-form (Ex)
4. Representing homology classes by embedded 2-spheres (Ex)

1.3 Lecture 3: Stable diffeomorphism and the Q-form Conjecture

1. Degree one normal maps to the 3-sphere (Ex)
2. Normal_1-types_of_4-manifolds_(Ex)
3. Freedman's classification and the Q-form hypothesis (Ex)

1.4 Lecture 4: The surgery machine applied in low dimensions

1. Integral homology 3-spheres embed in the 4-sphere (Ex)
2. Uniqueness of contractible coboundary (Ex)

2 The (stable) Cannon Conjecture

2.1 Lecture 1: An introduction to 3-manifolds

1. Betti numbers of 3-manifolds (Ex) solved
2. Non-prime solvable fundamental groups (Ex)
3. Atoroidal 3-manifolds (Ex)
4. Three dimensional Heisenberg group (Ex) solved
5. Circle actions on 3-manifolds (Ex)

2.2 Lecture 2: An introduction to hyperbolic groups

1. Torsion-free solvable hyperbolic groups (Ex)
2. Fundamental groups of surfaces (Ex)
3. Minimal dimension of BG (Ex)
4. Extensions of groups (Ex)
5. Boundaries of Fuchsian groups (Ex)

2.3 Lecture 3: Topological rigidity

1. Euler characteristic as surgery obstruction (Ex)
2. Borel Conjecture for the 2-torus (Ex)
3. Farrell-Jones Conjecture for finite groups (Ex)
4. Computation of certain L-groups I (Ex)
5. Computation of certain L-groups II (Ex)

2.4 Lecture 4: L²-invariants

1. L2-Betti numbers for the universal covering of the circle (Ex)
2. Atiyah Conjecture and finite groups (Ex)
3. Volume of a closed hyperbolic 3-manifold (Ex)
4. Thurston norm and the dual Thurston polytope (Ex)
5. Dual Thurston polytope of the 3-torus (Ex)

2.5 Lecture 5: The (stable) Cannon Conjecture

1. Closed manifolds are closed ANR-homology manifolds (Ex)
2. Products of ANR-homology manifolds (Ex)
3. Product rigidity (Ex)
4. Double suspension (Ex)
5. Surface groups as subgroups of hyperbolic groups (Ex)

3 Invariants of knots from Heegaard Floer homology

3.1 Lecture 1: Heegaard diagrams

1. Simple closed curves in surfaces (Ex)

3.2 Lecture 2: Floer homology

1. Elementary invariants of Heegaard diagrams (Ex)