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Algebraic Topology

Description: Algebraic Topology Quiz
Number of Questions: 14
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Tags: algebraic topology topology mathematics
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What is the fundamental group of a circle?

  1. $\mathbb{Z}$

  2. $\mathbb{R}$

  3. $\mathbb{Q}$

  4. $\mathbb{C}$

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Correct Option: A
Explanation:

The fundamental group of a circle is the group of homotopy classes of loops based at a point on the circle. This group is isomorphic to the group of integers, $\mathbb{Z}$, under the operation of addition.

What is the homology group of a sphere?

  1. $\mathbb{Z}$

  2. $\mathbb{R}$

  3. $\mathbb{Q}$

  4. $\mathbb{C}$

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Correct Option: A
Explanation:

The homology group of a sphere is the group of homology classes of closed loops on the sphere. This group is isomorphic to the group of integers, $\mathbb{Z}$, under the operation of addition.

What is the cohomology group of a torus?

  1. $\mathbb{Z}^2$

  2. $\mathbb{R}^2$

  3. $\mathbb{Q}^2$

  4. $\mathbb{C}^2$

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Correct Option: A
Explanation:

The cohomology group of a torus is the group of cohomology classes of closed loops on the torus. This group is isomorphic to the group of integers, $\mathbb{Z}^2$, under the operation of addition.

What is the Euler characteristic of a sphere?

  1. 1

  2. 2

  3. 3

  4. 4

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Correct Option: B
Explanation:

The Euler characteristic of a sphere is the alternating sum of the number of vertices, edges, and faces of the sphere. For a sphere, this sum is 2.

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.

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Correct Option: A
Explanation:

The Poincaré duality theorem states that the homology and cohomology groups of a manifold are isomorphic.

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.

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Correct Option: C
Explanation:

The Künneth theorem states that the homology and cohomology groups of a product space are the tensor products of the homology and cohomology groups of the individual spaces.

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.

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Correct Option: A
Explanation:

The Hurewicz theorem states that the homology groups of a space are isomorphic to the homology groups of its fundamental group.

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.

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Correct Option: A
Explanation:

The Whitehead theorem states that the fundamental group of a space is isomorphic to the homology group of its universal cover.

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.

Show answer
Correct Option: A
Explanation:

The Seifert-van Kampen theorem states that the fundamental group of a space is the free product of the fundamental groups of its path components.

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.

Show answer
Correct Option: A
Explanation:

The Alexander duality theorem states that the homology groups of a compact, orientable manifold are isomorphic to the cohomology groups of its dual manifold.

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.

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Correct Option: A
Explanation:

The Lefschetz duality theorem states that the homology groups of a compact, orientable manifold are isomorphic to the cohomology groups of its dual manifold.

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.

Show answer
Correct Option: A
Explanation:

The de Rham cohomology theorem states that the cohomology groups of a smooth manifold are isomorphic to the de Rham cohomology groups of the manifold.

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.

Show answer
Correct Option: A
Explanation:

The Hodge decomposition theorem states that the de Rham cohomology groups of a compact, orientable manifold can be decomposed into a direct sum of harmonic forms.

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.

Show answer
Correct Option: A
Explanation:

The Gauss-Bonnet theorem states that the integral of the Gaussian curvature of a compact, orientable surface is equal to 2\pi times the Euler characteristic of the surface.

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