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Homotopy Theory

Description: This quiz covers the fundamental concepts and theorems of Homotopy Theory, a branch of mathematics that studies the topological properties of spaces through continuous deformations.
Number of Questions: 15
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Tags: homotopy topology algebraic topology
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What is the fundamental group of a space X, denoted by (\pi_1(X))?

  1. The group of all continuous loops in X based at a fixed point.

  2. The group of all continuous maps from the unit circle to X.

  3. The group of all homotopy classes of continuous maps from the unit circle to X.

  4. The group of all continuous maps from X to the unit circle.

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Correct Option: C
Explanation:

The fundamental group (\pi_1(X)) is defined as the group of all homotopy classes of continuous maps from the unit circle (S^1) to X, where two maps are homotopic if they can be continuously deformed into each other while keeping the endpoints fixed.

What is the homology group of a space X, denoted by (H_n(X))?

  1. The group of all singular n-simplices in X.

  2. The group of all continuous maps from the n-sphere to X.

  3. The group of all homotopy classes of continuous maps from the n-sphere to X.

  4. The group of all singular n-chains in X.

Show answer
Correct Option: D
Explanation:

The homology group (H_n(X)) is defined as the group of all singular n-chains in X, modulo the group of all singular n-boundaries in X. A singular n-chain is a formal sum of n-simplices in X, and a singular n-boundary is a boundary of an (n+1)-simplex in X.

What is the cohomology group of a space X, denoted by (H^n(X))?

  1. The group of all continuous maps from X to the n-sphere.

  2. The group of all homotopy classes of continuous maps from X to the n-sphere.

  3. The group of all singular n-cochains in X.

  4. The group of all singular n-cocycles in X.

Show answer
Correct Option: D
Explanation:

The cohomology group (H^n(X)) is defined as the group of all singular n-cocycles in X, modulo the group of all singular n-coboundaries in X. A singular n-cochain is a function that assigns an integer to each n-simplex in X, and a singular n-coboundary is the coboundary of an (n-1)-cochain.

What is the Hurewicz theorem?

  1. It relates the homology groups of a space to its homotopy groups.

  2. It relates the cohomology groups of a space to its homotopy groups.

  3. It relates the homology groups of a space to its cohomology groups.

  4. It relates the homotopy groups of a space to its cohomology groups.

Show answer
Correct Option: A
Explanation:

The Hurewicz theorem states that for a simply connected space X, there is an isomorphism between the homology groups (H_n(X)) and the homotopy groups (\pi_n(X)) for (n \geq 2).

What is the Whitehead theorem?

  1. It states that every homotopy equivalence is a homology equivalence.

  2. It states that every homology equivalence is a homotopy equivalence.

  3. It states that every homotopy equivalence is a cohomology equivalence.

  4. It states that every cohomology equivalence is a homotopy equivalence.

Show answer
Correct Option: A
Explanation:

The Whitehead theorem states that if X and Y are two spaces and (f: X \to Y) is a homotopy equivalence, then (f) induces an isomorphism between the homology groups (H_n(X)) and (H_n(Y)) for all (n).

What is the Poincare duality theorem?

  1. It relates the homology groups of a manifold to its cohomology groups.

  2. It relates the cohomology groups of a manifold to its homology groups.

  3. It relates the homology groups of a manifold to its homotopy groups.

  4. It relates the homotopy groups of a manifold to its cohomology groups.

Show answer
Correct Option: A
Explanation:

The Poincare duality theorem states that for a closed, oriented n-manifold M, there is an isomorphism between the homology group (H_n(M)) and the cohomology group (H^{n-n}(M)).

What is the homology suspension theorem?

  1. It states that the homology groups of a space X are isomorphic to the homology groups of its suspension (\Sigma X).

  2. It states that the cohomology groups of a space X are isomorphic to the cohomology groups of its suspension (\Sigma X).

  3. It states that the homology groups of a space X are isomorphic to the homotopy groups of its suspension (\Sigma X).

  4. It states that the homotopy groups of a space X are isomorphic to the cohomology groups of its suspension (\Sigma X).

Show answer
Correct Option: A
Explanation:

The homology suspension theorem states that for a space X, there is an isomorphism between the homology groups (H_n(X)) and the homology groups (H_{n+1}(\Sigma X)) for all (n).

What is the Eilenberg-Steenrod axioms?

  1. It is a set of axioms that characterize the homology groups of a space.

  2. It is a set of axioms that characterize the cohomology groups of a space.

  3. It is a set of axioms that characterize the homotopy groups of a space.

  4. It is a set of axioms that characterize the homology and cohomology groups of a space.

Show answer
Correct Option: D
Explanation:

The Eilenberg-Steenrod axioms are a set of axioms that characterize the homology and cohomology groups of a space. They provide a foundation for the study of homology and cohomology theory.

What is the Kunneth theorem?

  1. It relates the homology groups of a product space to the homology groups of its factors.

  2. It relates the cohomology groups of a product space to the cohomology groups of its factors.

  3. It relates the homology groups of a product space to the homotopy groups of its factors.

  4. It relates the homotopy groups of a product space to the cohomology groups of its factors.

Show answer
Correct Option: A
Explanation:

The Kunneth theorem states that for two spaces X and Y, there is an isomorphism between the homology groups (H_n(X \times Y)) and the direct sum (\bigoplus_{i+j=n} H_i(X) \otimes H_j(Y)) for all (n).

What is the Lefschetz duality theorem?

  1. It relates the homology groups of a compact manifold to its cohomology groups.

  2. It relates the cohomology groups of a compact manifold to its homology groups.

  3. It relates the homology groups of a compact manifold to its homotopy groups.

  4. It relates the homotopy groups of a compact manifold to its cohomology groups.

Show answer
Correct Option: A
Explanation:

The Lefschetz duality theorem states that for a compact, oriented n-manifold M, there is an isomorphism between the homology group (H_n(M)) and the cohomology group (H^{n-n}(M)).

What is the Alexander duality theorem?

  1. It relates the homology groups of a compact, connected, orientable 3-manifold to its cohomology groups.

  2. It relates the cohomology groups of a compact, connected, orientable 3-manifold to its homology groups.

  3. It relates the homology groups of a compact, connected, orientable 3-manifold to its homotopy groups.

  4. It relates the homotopy groups of a compact, connected, orientable 3-manifold to its cohomology groups.

Show answer
Correct Option: A
Explanation:

The Alexander duality theorem states that for a compact, connected, orientable 3-manifold M, there is an isomorphism between the homology group (H_1(M)) and the cohomology group (H^2(M)), and an isomorphism between the homology group (H_2(M)) and the cohomology group (H^1(M)).

What is the homology sphere?

  1. A sphere that is homeomorphic to the n-sphere (S^n).

  2. A sphere that is homotopy equivalent to the n-sphere (S^n).

  3. A sphere that is homology equivalent to the n-sphere (S^n).

  4. A sphere that is cohomology equivalent to the n-sphere (S^n).

Show answer
Correct Option: C
Explanation:

A homology sphere is a sphere that is homology equivalent to the n-sphere (S^n). This means that there is an isomorphism between the homology groups of the sphere and the homology groups of (S^n).

What is the Poincare conjecture?

  1. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

  2. Every simply connected, closed 3-manifold is homotopy equivalent to the 3-sphere.

  3. Every simply connected, closed 3-manifold is homology equivalent to the 3-sphere.

  4. Every simply connected, closed 3-manifold is cohomology equivalent to the 3-sphere.

Show answer
Correct Option: A
Explanation:

The Poincare conjecture states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was proven by Grigori Perelman in 2002.

What is the sphere theorem?

  1. Every homotopy sphere is homeomorphic to a sphere.

  2. Every homotopy sphere is homology equivalent to a sphere.

  3. Every homotopy sphere is cohomology equivalent to a sphere.

  4. Every homotopy sphere is simply connected.

Show answer
Correct Option: A
Explanation:

The sphere theorem states that every homotopy sphere is homeomorphic to a sphere. This theorem was proven by Stephen Smale in 1961.

What is the Freudenthal suspension theorem?

  1. The suspension of a pointed space is a simply connected space.

  2. The suspension of a pointed space is a homology sphere.

  3. The suspension of a pointed space is a homotopy sphere.

  4. The suspension of a pointed space is a cohomology sphere.

Show answer
Correct Option: A
Explanation:

The Freudenthal suspension theorem states that the suspension of a pointed space is a simply connected space. This theorem was proven by Hans Freudenthal in 1937.

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