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Geometric Topology

Geometric Topology Quiz

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the fundamental group of a torus?

  1. $Z^2$
  2. $Z$
  3. $S^1$
  4. $Z imes Z$
Question 2 Multiple Choice (Single Answer)

What is the Poincaré duality theorem?

  1. A theorem relating the homology and cohomology of a manifold.
  2. A theorem relating the fundamental group and the homology of a manifold.
  3. A theorem relating the Euler characteristic and the genus of a surface.
  4. A theorem relating the curvature and the topology of a manifold.
Question 3 Multiple Choice (Single Answer)

What is a knot?

  1. A closed curve in 3-space.
  2. A closed surface in 3-space.
  3. A closed 3-manifold.
  4. A closed 4-manifold.
Question 4 Multiple Choice (Single Answer)

What is the Alexander polynomial of a knot?

  1. A polynomial that is associated to a knot and is an invariant of the knot.
  2. A polynomial that is associated to a knot and is not an invariant of the knot.
  3. A polynomial that is associated to a knot and is not an invariant of the knot.
  4. A polynomial that is associated to a knot and is not an invariant of the knot.
Question 5 Multiple Choice (Single Answer)

What is the Thurston geometrization conjecture?

  1. A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
  2. A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
  3. A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
  4. A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
Question 6 Multiple Choice (Single Answer)

What is the Dehn surgery theorem?

  1. A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
  2. A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
  3. A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
  4. A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
Question 7 Multiple Choice (Single Answer)

What is the Casson invariant?

  1. An invariant of a 3-manifold that is defined using Heegaard splittings.
  2. An invariant of a 3-manifold that is defined using Heegaard splittings.
  3. An invariant of a 3-manifold that is defined using Heegaard splittings.
  4. An invariant of a 3-manifold that is defined using Heegaard splittings.
Question 8 Multiple Choice (Single Answer)

What is the Floer homology of a 3-manifold?

  1. A homology theory for 3-manifolds that is defined using Morse theory.
  2. A homology theory for 3-manifolds that is defined using Morse theory.
  3. A homology theory for 3-manifolds that is defined using Morse theory.
  4. A homology theory for 3-manifolds that is defined using Morse theory.
Question 9 Multiple Choice (Single Answer)

What is the Seiberg-Witten invariant?

  1. An invariant of a 4-manifold that is defined using gauge theory.
  2. An invariant of a 4-manifold that is defined using gauge theory.
  3. An invariant of a 4-manifold that is defined using gauge theory.
  4. An invariant of a 4-manifold that is defined using gauge theory.
Question 10 Multiple Choice (Single Answer)

What is the Donaldson polynomial?

  1. An invariant of a 4-manifold that is defined using Donaldson theory.
  2. An invariant of a 4-manifold that is defined using Donaldson theory.
  3. An invariant of a 4-manifold that is defined using Donaldson theory.
  4. An invariant of a 4-manifold that is defined using Donaldson theory.
Question 11 Multiple Choice (Single Answer)

What is the Gromov-Witten invariant?

  1. An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
  2. An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
  3. An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
  4. An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
Question 12 Multiple Choice (Single Answer)

What is the Khovanov homology of a link?

  1. A homology theory for links that is defined using categorification.
  2. A homology theory for links that is defined using categorification.
  3. A homology theory for links that is defined using categorification.
  4. A homology theory for links that is defined using categorification.
Question 13 Multiple Choice (Single Answer)

What is the Rasmussen invariant?

  1. An invariant of a knot that is defined using Khovanov homology.
  2. An invariant of a knot that is defined using Khovanov homology.
  3. An invariant of a knot that is defined using Khovanov homology.
  4. An invariant of a knot that is defined using Khovanov homology.
Question 14 Multiple Choice (Single Answer)

What is the Heegaard Floer homology of a 3-manifold?

  1. A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
  2. A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
  3. A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
  4. A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
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