Geometric Topology
| Description: Geometric Topology Quiz | |
| Number of Questions: 14 | |
| Created by: Aliensbrain Bot | |
| Tags: topology manifolds knot theory 3-manifolds geometric group theory |
What is the fundamental group of a torus?
-
$Z^2$
-
$Z$
-
$S^1$
-
$Z imes Z$
The fundamental group of a torus is $Z imes Z$, which is generated by two loops that wind around the torus in different directions.
What is the Poincaré duality theorem?
-
A theorem relating the homology and cohomology of a manifold.
-
A theorem relating the fundamental group and the homology of a manifold.
-
A theorem relating the Euler characteristic and the genus of a surface.
-
A theorem relating the curvature and the topology of a manifold.
The Poincaré duality theorem states that the homology and cohomology of a manifold are isomorphic, which provides a powerful tool for studying the topology of manifolds.
What is a knot?
-
A closed curve in 3-space.
-
A closed surface in 3-space.
-
A closed 3-manifold.
-
A closed 4-manifold.
A knot is a closed curve in 3-space that does not intersect itself.
What is the Alexander polynomial of a knot?
-
A polynomial that is associated to a knot and is an invariant of the knot.
-
A polynomial that is associated to a knot and is not an invariant of the knot.
-
A polynomial that is associated to a knot and is not an invariant of the knot.
-
A polynomial that is associated to a knot and is not an invariant of the knot.
The Alexander polynomial of a knot is a polynomial that is associated to the knot and is an invariant of the knot, which means that it does not change if the knot is deformed.
What is the Thurston geometrization conjecture?
-
A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
-
A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
-
A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
-
A conjecture that states that every 3-manifold can be decomposed into a collection of simpler pieces.
The Thurston geometrization conjecture states that every 3-manifold can be decomposed into a collection of simpler pieces, called geometric pieces, which are either hyperbolic, spherical, or Euclidean.
What is the Dehn surgery theorem?
-
A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
-
A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
-
A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
-
A theorem that states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere.
The Dehn surgery theorem states that every 3-manifold can be obtained by performing Dehn surgery on a link in a 3-sphere, which involves cutting the 3-sphere along the link and then gluing it back together in a different way.
What is the Casson invariant?
-
An invariant of a 3-manifold that is defined using Heegaard splittings.
-
An invariant of a 3-manifold that is defined using Heegaard splittings.
-
An invariant of a 3-manifold that is defined using Heegaard splittings.
-
An invariant of a 3-manifold that is defined using Heegaard splittings.
The Casson invariant is an invariant of a 3-manifold that is defined using Heegaard splittings, which are a way of decomposing a 3-manifold into two handlebodies.
What is the Floer homology of a 3-manifold?
-
A homology theory for 3-manifolds that is defined using Morse theory.
-
A homology theory for 3-manifolds that is defined using Morse theory.
-
A homology theory for 3-manifolds that is defined using Morse theory.
-
A homology theory for 3-manifolds that is defined using Morse theory.
The Floer homology of a 3-manifold is a homology theory for 3-manifolds that is defined using Morse theory, which is a technique for studying the topology of a manifold by analyzing its critical points.
What is the Seiberg-Witten invariant?
-
An invariant of a 4-manifold that is defined using gauge theory.
-
An invariant of a 4-manifold that is defined using gauge theory.
-
An invariant of a 4-manifold that is defined using gauge theory.
-
An invariant of a 4-manifold that is defined using gauge theory.
The Seiberg-Witten invariant is an invariant of a 4-manifold that is defined using gauge theory, which is a mathematical framework for studying the interactions of elementary particles.
What is the Donaldson polynomial?
-
An invariant of a 4-manifold that is defined using Donaldson theory.
-
An invariant of a 4-manifold that is defined using Donaldson theory.
-
An invariant of a 4-manifold that is defined using Donaldson theory.
-
An invariant of a 4-manifold that is defined using Donaldson theory.
The Donaldson polynomial is an invariant of a 4-manifold that is defined using Donaldson theory, which is a mathematical framework for studying the topology of 4-manifolds.
What is the Gromov-Witten invariant?
-
An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
-
An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
-
An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
-
An invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory.
The Gromov-Witten invariant is an invariant of a symplectic 4-manifold that is defined using Gromov-Witten theory, which is a mathematical framework for studying the topology of symplectic manifolds.
What is the Khovanov homology of a link?
-
A homology theory for links that is defined using categorification.
-
A homology theory for links that is defined using categorification.
-
A homology theory for links that is defined using categorification.
-
A homology theory for links that is defined using categorification.
The Khovanov homology of a link is a homology theory for links that is defined using categorification, which is a mathematical framework for studying algebraic structures by replacing them with equivalent categories.
What is the Rasmussen invariant?
-
An invariant of a knot that is defined using Khovanov homology.
-
An invariant of a knot that is defined using Khovanov homology.
-
An invariant of a knot that is defined using Khovanov homology.
-
An invariant of a knot that is defined using Khovanov homology.
The Rasmussen invariant is an invariant of a knot that is defined using Khovanov homology, which is a homology theory for links that is defined using categorification.
What is the Heegaard Floer homology of a 3-manifold?
-
A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
-
A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
-
A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
-
A homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory.
The Heegaard Floer homology of a 3-manifold is a homology theory for 3-manifolds that is defined using Heegaard splittings and Floer theory, which is a mathematical framework for studying the topology of 3-manifolds.