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 | |
| Created by: Aliensbrain Bot | |
| Tags: homotopy topology algebraic topology |
What is the fundamental group of a space X, denoted by (\pi_1(X))?
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The group of all continuous loops in X based at a fixed point.
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The group of all continuous maps from the unit circle to X.
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The group of all homotopy classes of continuous maps from the unit circle to X.
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The group of all continuous maps from X to the unit circle.
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))?
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The group of all singular n-simplices in X.
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The group of all continuous maps from the n-sphere to X.
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The group of all homotopy classes of continuous maps from the n-sphere to X.
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The group of all singular n-chains in X.
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))?
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The group of all continuous maps from X to the n-sphere.
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The group of all homotopy classes of continuous maps from X to the n-sphere.
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The group of all singular n-cochains in X.
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The group of all singular n-cocycles in X.
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?
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It relates the homology groups of a space to its homotopy groups.
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It relates the cohomology groups of a space to its homotopy groups.
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It relates the homology groups of a space to its cohomology groups.
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It relates the homotopy groups of a space to its cohomology groups.
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?
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It states that every homotopy equivalence is a homology equivalence.
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It states that every homology equivalence is a homotopy equivalence.
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It states that every homotopy equivalence is a cohomology equivalence.
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It states that every cohomology equivalence is a homotopy equivalence.
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?
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It relates the homology groups of a manifold to its cohomology groups.
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It relates the cohomology groups of a manifold to its homology groups.
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It relates the homology groups of a manifold to its homotopy groups.
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It relates the homotopy groups of a manifold to its cohomology groups.
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?
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It states that the homology groups of a space X are isomorphic to the homology groups of its suspension (\Sigma X).
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It states that the cohomology groups of a space X are isomorphic to the cohomology groups of its suspension (\Sigma X).
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It states that the homology groups of a space X are isomorphic to the homotopy groups of its suspension (\Sigma X).
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It states that the homotopy groups of a space X are isomorphic to the cohomology groups of its suspension (\Sigma X).
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?
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It is a set of axioms that characterize the homology groups of a space.
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It is a set of axioms that characterize the cohomology groups of a space.
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It is a set of axioms that characterize the homotopy groups of a space.
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It is a set of axioms that characterize the homology and cohomology groups of a space.
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?
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It relates the homology groups of a product space to the homology groups of its factors.
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It relates the cohomology groups of a product space to the cohomology groups of its factors.
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It relates the homology groups of a product space to the homotopy groups of its factors.
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It relates the homotopy groups of a product space to the cohomology groups of its factors.
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?
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It relates the homology groups of a compact manifold to its cohomology groups.
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It relates the cohomology groups of a compact manifold to its homology groups.
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It relates the homology groups of a compact manifold to its homotopy groups.
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It relates the homotopy groups of a compact manifold to its cohomology groups.
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?
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It relates the homology groups of a compact, connected, orientable 3-manifold to its cohomology groups.
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It relates the cohomology groups of a compact, connected, orientable 3-manifold to its homology groups.
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It relates the homology groups of a compact, connected, orientable 3-manifold to its homotopy groups.
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It relates the homotopy groups of a compact, connected, orientable 3-manifold to its cohomology groups.
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?
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A sphere that is homeomorphic to the n-sphere (S^n).
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A sphere that is homotopy equivalent to the n-sphere (S^n).
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A sphere that is homology equivalent to the n-sphere (S^n).
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A sphere that is cohomology equivalent to the n-sphere (S^n).
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?
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Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
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Every simply connected, closed 3-manifold is homotopy equivalent to the 3-sphere.
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Every simply connected, closed 3-manifold is homology equivalent to the 3-sphere.
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Every simply connected, closed 3-manifold is cohomology equivalent to the 3-sphere.
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?
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Every homotopy sphere is homeomorphic to a sphere.
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Every homotopy sphere is homology equivalent to a sphere.
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Every homotopy sphere is cohomology equivalent to a sphere.
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Every homotopy sphere is simply connected.
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?
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The suspension of a pointed space is a simply connected space.
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The suspension of a pointed space is a homology sphere.
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The suspension of a pointed space is a homotopy sphere.
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The suspension of a pointed space is a cohomology sphere.
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.