Geometric Topology Quizzes: Master the Fundamentals of Geometric Objects

Question 1

Define a closed curve in 3-space that does not intersect itself.

Accepted Answer

A loop

Answer

A path

Answer

A circle

Answer

A line

Question 2

Identify which of the following is not a property of a manifold.

Accepted Answer

Non-orientable

Answer

Locally Euclidean

Answer

Connected

Answer

Compact

Question 3

Define the Gauss linking number of two unlinked circles.

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 4

Which fundamental concept is not covered in geometric topology?

Accepted Answer

Matrices

Answer

Knots

Answer

Links

Answer

Manifolds

Question 5

Which characteristic of a knot is a topological invariant?

Accepted Answer

Knot type

Answer

Knot length

Answer

Knot orientation

Answer

Knot complexity

Question 6

Which statement best describes vectors in the context of geometric topology?

Accepted Answer

Vectors are sometimes used as a tool in geometric topology.

Answer

Vectors are included as a fundamental concept in geometric topology.

Answer

Vectors are not a fundamental concept in geometric topology.

Answer

Vectors are essential for understanding geometric topology.

Question 7

What is the topological invariant used to classify knots?

Accepted Answer

Knot group

Answer

Genus

Answer

Linking number

Answer

Euler characteristic

Question 8

For a non-orientable surface, what is its Euler characteristic?

Accepted Answer

Undefined

Answer

0

Answer

1

Answer

2

Question 9

Which of the following is NOT a 3-manifold?

Accepted Answer

Möbius strip

Answer

Sphere

Answer

Torus

Answer

Klein bottle

Question 10

How are the homology group and fundamental group of a manifold related?

Accepted Answer

The homology group is a quotient of the fundamental group.

Answer

The fundamental group is a subgroup of the homology group.

Answer

They are isomorphic.

Answer

They are unrelated.

Question 11

Which of the following is a closed, non-intersecting curve in 3-space?

Accepted Answer

Knot

Answer

Sphere

Answer

Torus

Answer

Möbius strip

Question 12

Which of the following is NOT a fundamental concept in geometric topology?

Accepted Answer

Graphs

Answer

Knots

Answer

Links

Answer

Manifolds

Question 13

Define a homology group, describing its significance in geometric topology.

Accepted Answer

A group associated with a topological space, capturing its 'holes' and describing its topological structure.

Answer

A set of all closed curves in a space, providing information about its local geometry.

Answer

A group that defines the topology of a surface, enabling the classification of different surfaces.

Answer

A measure of the curvature of a manifold, indicating its intrinsic geometric properties.

Question 14

Which of the following is NOT a major challenge in contemporary geometric topology?

Accepted Answer

Solving the Riemann hypothesis, a famous unsolved problem in number theory.

Answer

Classification of 3-manifolds, determining their topological properties and relationships.

Answer

Understanding the topology of knots and links, exploring their mathematical structure and applications.

Answer

Developing new topological invariants, creating tools to distinguish and classify topological objects.

Question 15

Name the theorem that establishes a connection between knots and braids.

Accepted Answer

Wirtinger's Theorem, representing every knot as a closed braid.

Answer

Gauss's Linking Theorem, describing the linking number of two closed curves.

Answer

Alexander's Theorem, relating the knot group to the homology of its complement.

Answer

Reidemeister's Theorem, providing a set of moves to simplify and classify knots.

Question 16

Define a topological invariant of a link and provide an example.

Accepted Answer

A numerical or algebraic property that remains unchanged under continuous deformations of the link, such as its linking number.

Answer

A geometric property that describes the shape and orientation of the link, such as its genus.

Answer

A group-theoretic property that characterizes the fundamental group of the link's complement.

Answer

A measure of the total length of the link's components.

Question 17

Determine the fundamental group of a closed, orientable surface.

Accepted Answer

A free group on two generators, representing the non-trivial loops on the surface.

Answer

A finite group, indicating a surface with a finite number of loops.

Answer

A cyclic group, indicating a surface with a single loop.

Answer

An infinite group, indicating a surface with an infinite number of loops.

Question 18

What is the Betti number of a closed, orientable surface of genus g?

Accepted Answer

2g

Answer

0

Answer

1

Answer

Infinite

Question 19

Which of the following is NOT a fundamental concept in geometric topology?

Accepted Answer

Categories

Answer

Knots

Answer

Groups

Answer

Manifolds

Question 20

A knot is a:

Accepted Answer

Closed curve in 3-space that does not intersect itself.

Answer

Surface with no boundary

Answer

Group with a single generator

Answer

Manifold with a single boundary

Question 21

A manifold is a topological space that:

Accepted Answer

Is locally Euclidean

Answer

Has no boundary

Answer

Is orientable

Answer

Is compact

Question 22

The fundamental group of a surface of genus g is:

Accepted Answer

A free group with 2g generators

Answer

A group with one generator

Answer

A group with two generators

Answer

A cyclic group

Question 23

A triangulation of a manifold is:

Accepted Answer

A decomposition into simplices

Answer

A covering by open sets

Answer

A partition into disjoint subsets

Answer

A homeomorphism to a Euclidean space

Question 24

Which of the following characteristic properties of knots is related to whether it can be continuously deformed into an unknot?

Accepted Answer

Unknottedness

Answer

Triviality

Answer

Genus

Answer

Euler characteristic

Question 25

The fundamental group of a closed, orientable surface of genus g is isomorphic to:

Accepted Answer

Z/gZ

Answer

Z

Answer

D_g

Answer

S_g

Question 26

The degree of a map between two manifolds is equal to:

Accepted Answer

The number of preimages of a regular value

Answer

The number of critical values

Answer

The Euler characteristic of the domain manifold

Answer

The genus of the codomain manifold

Question 27

The fundamental group of the complement of a knot is:

Accepted Answer

Free

Answer

Abelian

Answer

Cyclic

Answer

Trivial

Question 28

Which of the following is a fundamental topological invariant of knots?

Accepted Answer

Knot polynomial

Answer

Knot volume

Answer

Knot area

Answer

Knot length

Question 29

In a link diagram, what does a positive crossing indicate?

Accepted Answer

The strand passes over another strand

Answer

The strands are parallel

Answer

The strand passes under another strand

Answer

The strands are perpendicular

Question 30

What is the dimension of a compact orientable 3-manifold?

Accepted Answer

3

Answer

2

Answer

4

Question 31

Which theorem states that every compact surface without boundary can be triangulated?

Accepted Answer

Whitney triangulation theorem

Answer

Poincaré duality theorem

Answer

Gauss-Bonnet theorem

Answer

Jordan curve theorem

Question 32

What is the fundamental group of a torus?

Accepted Answer

Z × Z

Answer

S<sub>3</sub>

Answer

Z<sub>2</sub>

Answer

Z

Question 33

What is the Dehn surgery on a knot K along a longitude?

Accepted Answer

Produces a torus

Answer

Produces a sphere

Answer

Produces a Klein bottle

Answer

Produces a projective plane

Question 34

The homology group of a manifold consists of:

Accepted Answer

Abelian groups.

Answer

Permutation groups.

Answer

Linear spaces.

Answer

Topological spaces.

Question 35

Which of the following is a topological invariant of a knot?

Accepted Answer

Alexander polynomial.

Answer

Length.

Answer

Area.

Answer

Volume.

Question 36

What is the dimension of a manifold?

Accepted Answer

An integer greater than or equal to 1.

Answer

A complex number.

Answer

A rational number.

Question 37

The fundamental group of a space is a group whose elements are:

Accepted Answer

Homotopy classes of loops

Answer

Knots

Answer

Surfaces

Answer

Topological spaces

Question 38

Which statement is a direct consequence of the Poincaré duality theorem?

Accepted Answer

Every closed manifold is orientable.

Answer

Every smooth manifold is a differentiable manifold.

Answer

Any topological manifold is a smooth manifold.

Answer

Every compact manifold is a Riemannian manifold.

Question 39

Which of the following is **not** a topological invariant of a knot?

Accepted Answer

Length

Answer

Crossing number

Answer

Unknotting number

Answer

Genus

Question 40

Which of the following is a simply connected manifold?

Accepted Answer

S^2

Answer

T^2

Answer

Klein bottle

Answer

Möbius strip

Question 41

What is the Seifert-Van Kampen theorem used for?

Accepted Answer

Calculating the fundamental group of a space by combining known groups

Answer

Classifying knots

Answer

Calculating the homology groups of a space

Question 42

Which of the following is a homology sphere?

Accepted Answer

A 3-manifold with identical homology groups to a 3-sphere

Answer

A 3-manifold with an identical Euler characteristic to a 3-sphere

Answer

A 3-manifold with an identical fundamental group to a 3-sphere

Question 43

What is the Poincaré duality theorem used for?

Accepted Answer

Relating the homology and cohomology groups of a manifold

Answer

Classifying surfaces

Answer

Proving the four-color theorem

Answer

Calculating the fundamental group of a manifold

Question 44

What is the fundamental group of a sphere?

Accepted Answer

The trivial group.

Answer

A free group on two generators.

Answer

The cyclic group of order 2.

Answer

The fundamental group of a torus.

Question 45

Two knots are considered equivalent if they can be transformed into each other by:

Accepted Answer

Isotopy.

Answer

Homeomorphism.

Answer

Diffeomorphism.

Question 46

A manifold is a topological space that locally resembles:

Accepted Answer

Euclidean space.

Answer

A torus.

Answer

A sphere.

Answer

A projective plane.

Question 47

The connected sum of two tori is:

Accepted Answer

A surface with two holes.

Answer

A sphere.

Answer

A Klein bottle.

Answer

A projective plane.