Algebraic Topology
| Description: Algebraic Topology Quiz | |
| Number of Questions: 14 | |
| Created by: Aliensbrain Bot | |
| Tags: algebraic topology topology mathematics |
What is the fundamental group of a circle?
-
$\mathbb{Z}$
-
$\mathbb{R}$
-
$\mathbb{Q}$
-
$\mathbb{C}$
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?
-
$\mathbb{Z}$
-
$\mathbb{R}$
-
$\mathbb{Q}$
-
$\mathbb{C}$
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?
-
$\mathbb{Z}^2$
-
$\mathbb{R}^2$
-
$\mathbb{Q}^2$
-
$\mathbb{C}^2$
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
-
2
-
3
-
4
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?
-
A theorem that relates the homology and cohomology groups of a manifold.
-
A theorem that relates the fundamental group and homology groups of a manifold.
-
A theorem that relates the cohomology group and Euler characteristic of a manifold.
-
A theorem that relates the homology group and Euler characteristic of a manifold.
The Poincaré duality theorem states that the homology and cohomology groups of a manifold are isomorphic.
What is the Künneth theorem?
-
A theorem that relates the homology groups of two spaces.
-
A theorem that relates the cohomology groups of two spaces.
-
A theorem that relates the homology and cohomology groups of a product space.
-
A theorem that relates the fundamental group and homology groups of a product space.
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?
-
A theorem that relates the homology groups of a space to its fundamental group.
-
A theorem that relates the cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology and cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
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?
-
A theorem that relates the fundamental group of a space to its homology groups.
-
A theorem that relates the cohomology groups of a space to its homology groups.
-
A theorem that relates the homology and cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
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?
-
A theorem that relates the fundamental group of a space to its homology groups.
-
A theorem that relates the cohomology groups of a space to its homology groups.
-
A theorem that relates the homology and cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
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?
-
A theorem that relates the homology groups of a space to its cohomology groups.
-
A theorem that relates the cohomology groups of a space to its homology groups.
-
A theorem that relates the homology and cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
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?
-
A theorem that relates the homology groups of a space to its cohomology groups.
-
A theorem that relates the cohomology groups of a space to its homology groups.
-
A theorem that relates the homology and cohomology groups of a space to its fundamental group.
-
A theorem that relates the homology group and Euler characteristic of a space to its fundamental group.
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?
-
A theorem that relates the cohomology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology and cohomology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology group and Euler characteristic of a space to its de Rham cohomology groups.
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?
-
A theorem that relates the cohomology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology and cohomology groups of a space to its de Rham cohomology groups.
-
A theorem that relates the homology group and Euler characteristic of a space to its de Rham cohomology groups.
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?
-
A theorem that relates the curvature of a surface to its Euler characteristic.
-
A theorem that relates the curvature of a surface to its homology groups.
-
A theorem that relates the curvature of a surface to its cohomology groups.
-
A theorem that relates the curvature of a surface to its fundamental group.
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.