# 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.

## [edit] 1 Surgery: high-d methods in low-d

### [edit] 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)

### [edit] 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)

### [edit] 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)

### [edit] 1.4 Lecture 4: The surgery machine applied in low dimensions

### [edit] 1.5 Lecture 5: Topological concordance of classical knots: Where are we?

## [edit] 2 The (stable) Cannon Conjecture

### [edit] 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) solved
- Circle actions on 3-manifolds (Ex)

### [edit] 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)

### [edit] 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)

### [edit] 2.4 Lecture 4: L^{2}-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)

### [edit] 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)