MATRIX 2019 Interactions: Exercises
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Participants are encouraged to work on the exercises and contribute solutions on the discussion page. | Participants are encouraged to work on the exercises and contribute solutions on the discussion page. | ||
| − | == Surgery: high-d methods in low-d == | + | == Surgery: high-d methods in low-d== |
| − | === Lecture 1: Normal maps and the surgery obstruction === | + | === Lecture 1: Normal maps and the surgery obstruction=== |
# [[Stability of vector bundles (Ex)]] | # [[Stability of vector bundles (Ex)]] | ||
# [[Normal maps - (non)-examples (Ex)]] | # [[Normal maps - (non)-examples (Ex)]] | ||
| Line 11: | Line 11: | ||
# [[Surgery obstruction, Arf-invariant (Ex)]] | # [[Surgery obstruction, Arf-invariant (Ex)]] | ||
=== Lecture 2: Foundations of topological 4-manifolds === | === 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)]] | ||
| + | |||
=== Lecture 3: Stable diffeomorphism and the Q-form Conjecture === | === 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)]] | ||
=== Lecture 4: The surgery machine applied in low dimensions === | === Lecture 4: The surgery machine applied in low dimensions === | ||
| − | === Lecture 5: Topological concordance of classical | + | # [[Integral homology 3-spheres embed in the 4-sphere (Ex)]] |
| + | # [[Uniqueness of contractible coboundary (Ex)]] | ||
| + | |||
| + | === Lecture 5: Topological concordance of classical knots: Where are we?=== | ||
== The (stable) Cannon Conjecture == | == The (stable) Cannon Conjecture == | ||
=== Lecture 1: An introduction to 3-manifolds === | === Lecture 1: An introduction to 3-manifolds === | ||
| − | # [[Betti numbers of 3-manifolds (Ex)]] | + | # [[Betti numbers of 3-manifolds (Ex)]] solved |
# [[Non-prime solvable fundamental groups (Ex)]] | # [[Non-prime solvable fundamental groups (Ex)]] | ||
# [[Atoroidal 3-manifolds (Ex)]] | # [[Atoroidal 3-manifolds (Ex)]] | ||
| − | # [[Three dimensional Heisenberg group (Ex)]] | + | # [[Three dimensional Heisenberg group (Ex)]] solved |
# [[Circle actions on 3-manifolds (Ex)]] | # [[Circle actions on 3-manifolds (Ex)]] | ||
| Line 39: | Line 50: | ||
=== Lecture 4: L<sup>2</sup>-invariants === | === Lecture 4: L<sup>2</sup>-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)]] | ||
=== Lecture 5: The (stable) Cannon Conjecture === | === 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)]] | ||
== Invariants of knots from Heegaard Floer homology == | == Invariants of knots from Heegaard Floer homology == | ||
| Line 47: | Line 67: | ||
=== Lecture 2: Floer homology === | === Lecture 2: Floer homology === | ||
| + | # [[Elementary invariants of Heegaard diagrams (Ex)]] | ||
| + | |||
=== Lecture 3: Knot Floer homology === | === Lecture 3: Knot Floer homology === | ||
=== Lecture 4: The Upsilon invariant === | === Lecture 4: The Upsilon invariant === | ||
Latest revision as of 06:20, 10 January 2019
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: 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)
[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)
