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Algebraic Topology

Algebraic Topology Quiz

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the fundamental group of a circle?

  1. $\mathbb{Z}$
  2. $\mathbb{R}$
  3. $\mathbb{Q}$
  4. $\mathbb{C}$
Question 2 Multiple Choice (Single Answer)

What is the homology group of a sphere?

  1. $\mathbb{Z}$
  2. $\mathbb{R}$
  3. $\mathbb{Q}$
  4. $\mathbb{C}$
Question 3 Multiple Choice (Single Answer)

What is the cohomology group of a torus?

  1. $\mathbb{Z}^2$
  2. $\mathbb{R}^2$
  3. $\mathbb{Q}^2$
  4. $\mathbb{C}^2$
Question 4 Multiple Choice (Single Answer)

What is the Euler characteristic of a sphere?

  1. 1
  2. 2
  3. 3
  4. 4
Question 5 Multiple Choice (Single Answer)

What is the Poincaré duality theorem?

  1. A theorem that relates the homology and cohomology groups of a manifold.
  2. A theorem that relates the fundamental group and homology groups of a manifold.
  3. A theorem that relates the cohomology group and Euler characteristic of a manifold.
  4. A theorem that relates the homology group and Euler characteristic of a manifold.
Question 6 Multiple Choice (Single Answer)

What is the Künneth theorem?

  1. A theorem that relates the homology groups of two spaces.
  2. A theorem that relates the cohomology groups of two spaces.
  3. A theorem that relates the homology and cohomology groups of a product space.
  4. A theorem that relates the fundamental group and homology groups of a product space.
Question 7 Multiple Choice (Single Answer)

What is the Hurewicz theorem?

  1. A theorem that relates the homology groups of a space to its fundamental group.
  2. A theorem that relates the cohomology groups of a space to its fundamental group.
  3. A theorem that relates the homology and cohomology groups of a space to its fundamental group.
  4. A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
Question 8 Multiple Choice (Single Answer)

What is the Whitehead theorem?

  1. A theorem that relates the fundamental group of a space to its homology groups.
  2. A theorem that relates the cohomology groups of a space to its homology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its fundamental group.
  4. A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
Question 9 Multiple Choice (Single Answer)

What is the Seifert-van Kampen theorem?

  1. A theorem that relates the fundamental group of a space to its homology groups.
  2. A theorem that relates the cohomology groups of a space to its homology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its fundamental group.
  4. A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
Question 10 Multiple Choice (Single Answer)

What is the Alexander duality theorem?

  1. A theorem that relates the homology groups of a space to its cohomology groups.
  2. A theorem that relates the cohomology groups of a space to its homology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its fundamental group.
  4. A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
Question 11 Multiple Choice (Single Answer)

What is the Lefschetz duality theorem?

  1. A theorem that relates the homology groups of a space to its cohomology groups.
  2. A theorem that relates the cohomology groups of a space to its homology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its fundamental group.
  4. A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
Question 12 Multiple Choice (Single Answer)

What is the de Rham cohomology theorem?

  1. A theorem that relates the cohomology groups of a space to its de Rham cohomology groups.
  2. A theorem that relates the homology groups of a space to its de Rham cohomology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its de Rham cohomology groups.
  4. A theorem that relates the homology group and Euler characteristic of a space to its de Rham cohomology groups.
Question 13 Multiple Choice (Single Answer)

What is the Hodge decomposition theorem?

  1. A theorem that relates the cohomology groups of a space to its de Rham cohomology groups.
  2. A theorem that relates the homology groups of a space to its de Rham cohomology groups.
  3. A theorem that relates the homology and cohomology groups of a space to its de Rham cohomology groups.
  4. A theorem that relates the homology group and Euler characteristic of a space to its de Rham cohomology groups.
Question 14 Multiple Choice (Single Answer)

What is the Gauss-Bonnet theorem?

  1. A theorem that relates the curvature of a surface to its Euler characteristic.
  2. A theorem that relates the curvature of a surface to its homology groups.
  3. A theorem that relates the curvature of a surface to its cohomology groups.
  4. A theorem that relates the curvature of a surface to its fundamental group.
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