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.


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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[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

  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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 3 Invariants of knots from Heegaard Floer homology

[edit] 3.1 Lecture 1: Heegaard diagrams

  1. Simple closed curves in surfaces (Ex)

[edit] 3.2 Lecture 2: Floer homology

  1. Elementary invariants of Heegaard diagrams (Ex)

[edit] 3.3 Lecture 3: Knot Floer homology

[edit] 3.4 Lecture 4: The Upsilon invariant

[edit] 3.5 Lecture 5: Further applications

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