MATRIX 2019 Interactions: Exercises
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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
- Connected sum of topological manifolds (Ex)
- Quillen plus construction (Ex)
- Stability of the E8-form (Ex)
- Representing homology classes by embedded 2-spheres (Ex)
1.3 Lecture 3: Stable diffeomorphism and the Q-form Conjecture
- Degree one normal maps to the 3-sphere (Ex)
- Normal_1-types_of_4-manifolds_(Ex)
- Freedman's classification and the Q-form hypothesis (Ex)
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) solved
- 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)
2.5 Lecture 5: The (stable) Cannon Conjecture
- Closed manifolds are closed ANR-homology manifolds (Ex)
- Products of ANR-homology manifolds (Ex)
- Product rigidity (Ex)
- Double suspension (Ex)
- Surface groups as subgroups of hyperbolic groups (Ex)