MATRIX 2019 Interactions: Exercises
From Manifold Atlas
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
- Stability of vector bundles (Ex)
- Normal maps - (non)-examples (Ex)
- Immersing n-spheres in 2n-space (Ex)
- Surgery obstruction, signature (Ex)
- Surgery obstruction, Arf-invariant (Ex)
1.2 Lecture 2: Foundations of topological 4-manifolds
1.3 Lecture 3: Stable diffeomorphism and the Q-form Conjecture
1.4 Lecture 4: The surgery machine applied in low dimensions
1.5 Lecture 5: Topological concordance of classical knots: Where are we?
2 The (stable) Cannon Conjecture
2.1 Lecture 1: An introduction to 3-manifolds
- Betti numbers of 3-manifolds (Ex)
- Non-prime solvable fundamental groups (Ex)
- Atoroidal 3-manifolds (Ex)
- Three dimensional Heisenberg group (Ex)
- Circle actions on 3-manifolds (Ex)
2.2 Lecture 2: An introduction to hyperbolic groups
- Torsion-free solvable hyperbolic groups (Ex)
- Fundamental groups of surfaces (Ex)
- Minimal dimension of BG (Ex)
- Extensions of groups (Ex)
- Boundaries of Fuchsian groups (Ex)
2.3 Lecture 3: Topological rigidity
- Euler characteristic as surgery obstruction (Ex)
- Borel Conjecture for the 2-torus (Ex)
- Farrell-Jones Conjecture for finite groups (Ex)
- Computation of certain L-groups I (Ex)
- Computation of certain L-groups II (Ex)
2.4 Lecture 4: L2-invariants
- L2-Betti numbers for the universal covering of the circle (Ex)
- Atiyah Conjecture and finite groups (Ex)
- Volume of a closed hyperbolic 3-manifold (Ex)
- Thurston norm and the dual Thurston polytope (Ex)
- Dual Thurston polytope of the 3-torus (Ex)