Geometric Topology: Master the Interplay of Geometry and Topology

Question 1

Which of the following is a fundamental theorem in Geometric Topology that establishes a connection between the geometry and topology of a Riemannian manifold?

Accepted Answer

Gauss-Bonnet Theorem

Answer

Poincaré Duality Theorem

Answer

Dehn-Somerville Theorem

Answer

Stokes' Theorem

Question 2

What is the fundamental group of a simply connected space, a space with no holes or tunnels?

Accepted Answer

{e}

Answer

ℤ

Answer

S¹

Answer

Free group

Question 3

Which of the following concepts is NOT central to Geometric Topology?

Accepted Answer

Inverse function theorem

Answer

Manifolds

Answer

Covering spaces

Answer

Homology and Cohomology

Question 4

What is the defining characteristic of a simply connected manifold?

Accepted Answer

It has no non-contractible closed curves.

Answer

It is homeomorphic to a sphere.

Answer

It is orientable.

Answer

It is compact.

Question 5

In geometric topology, which core concept involves studying intersections of submanifolds and their impact on the topology of the entire manifold?

Accepted Answer

Intersection theory

Answer

Homology groups

Answer

Spectral sequences

Answer

Algebraic geometry

Question 6

Which of the following is not a type of manifold, a key object of study in geometric topology?

Accepted Answer

Topological ring

Answer

Surface

Answer

3-manifold

Answer

4-ball

Question 7

Explain the key differences between topological manifolds and smooth manifolds. How does this distinction affect their study in geometric topology?

Accepted Answer

Topological manifolds are defined by their topological properties, while smooth manifolds are defined by their geometric properties.

Accepted Answer

Smooth manifolds require their transition maps to be differentiable, while topological manifolds do not.

Accepted Answer

Topological manifolds can have singularities, while smooth manifolds do not.

Question 8

Which of the following properties does a **differentiable manifold** NOT possess?

Accepted Answer

Smooth surface

Answer

Locally Euclidean

Answer

Tangent space at each point

Answer

Well-defined notion of differential structure

Question 9

What is the essential characteristic of a **homology group**?

Accepted Answer

Associated with a chain complex

Answer

Comprised of permutations

Answer

Contains matrices

Answer

Consists of vectors

Question 10

Choose the correct definition of a **vector bundle**.

Accepted Answer

Fibre bundle with vector space fibres

Answer

Bundle of smooth manifolds

Answer

Bundle of topological spaces

Answer

Bundle of groups

Question 11

Which property characterizes a **smooth manifold**?

Accepted Answer

Locally Euclidean with well-defined tangent space

Answer

Locally compact with well-defined distance notion

Answer

Orientable with well-defined volume notion

Answer

Closed with well-defined boundary notion

Question 12

State the essence of the **Whitney embedding theorem**.

Accepted Answer

Every smooth manifold can be embedded in Euclidean space

Answer

Every smooth manifold is homeomorphic to a sphere

Answer

Every smooth manifold is orientable

Answer

Every smooth manifold is closed

Question 13

What is the significance of the Poincaré duality theorem?

Accepted Answer

It provides a connection between the homology and cohomology groups of a manifold, revealing relationships between different topological properties.

Answer

It provides a tool to determine if a manifold is simply connected, giving insights into its connectivity.

Answer

It offers a method to construct triangulations of manifolds, facilitating their geometric analysis.

Answer

It helps prove the existence of non-orientable surfaces, highlighting their unique topological properties.

Question 14

How is the fundamental group of a manifold related to its first homology group?

Accepted Answer

The relationship depends on the specific manifold, reflecting the complex interplay between algebra and topology.

Answer

They are always isomorphic, indicating a strong relationship between algebraic and topological properties.

Answer

They are always equal, implying that the fundamental group completely determines the homology group.

Answer

They are always different, highlighting the distinct nature of algebraic and topological information.

Question 15

Provide a geometric interpretation of the cup product in cohomology.

Accepted Answer

It corresponds to the intersection of submanifolds within the topological space, revealing relationships between geometric objects.

Answer

It represents the product of homology classes, providing an algebraic tool to study topological properties.

Answer

It facilitates the computation of the Euler characteristic, offering insights into the overall topology of the space.

Answer

It aids in the triangulation of manifolds, enabling the construction of simplicial complexes for geometric analysis.

Question 16

What is the Euler characteristic of a torus?

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 17

What is the fundamental group of a circle?

Accepted Answer

ℤ

Answer

ℤ/2ℤ

Answer

0

Answer

1

Question 18

Which of the following is a consequence of the Gauss-Bonnet Theorem?

Accepted Answer

The Euler characteristic of a compact surface is equal to its genus

Answer

The total curvature of a compact surface is zero

Answer

The fundamental group of a compact surface is finite

Answer

The genus of a compact surface is equal to its number of holes

Question 19

What is the intersection of a plane and a sphere?

Accepted Answer

A circle

Answer

A point

Answer

A line

Answer

A square

Question 20

Which of the following is a knot?

Accepted Answer

A closed curve in ℝ³ that does not intersect itself

Answer

A loop in a surface

Answer

A path in a graph

Answer

A straight line

Question 21

Identify the answer choice that does NOT represent a fundamental group of a closed orientable surface:

Accepted Answer

S3

Answer

Z

Answer

Z/2Z

Answer

Z x Z

Question 22

Identify the homology of a circle:

Accepted Answer

Z

Answer

Z/2Z

Answer

Z x Z

Answer

0

Question 23

Identify the orientable surface from the following options:

Accepted Answer

A projective plane

Answer

The real line

Answer

A Möbius strip

Answer

A Klein bottle

Question 24

Determine the Euler characteristic of a Klein bottle:

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 25

Which property is NOT characteristic of a compact manifold?

Accepted Answer

Uncountable

Answer

Closed and bounded

Answer

Metrizable

Answer

Hausdorff

Question 26

Which theorem establishes a relationship between the Euler characteristic of a triangulated manifold and its vertices, edges, and faces?

Accepted Answer

Dehn-Sommerville theorem

Answer

Poincaré duality theorem

Answer

Gauss-Bonnet theorem

Answer

Stokes' theorem

Question 27

What is the homology group of a circle?

Accepted Answer

Infinite cyclic group

Answer

Finite cyclic group

Answer

Trivial group

Answer

Free group of rank 1

Question 28

Define the signature of a manifold.

Accepted Answer

A topological invariant that distinguishes between different types of manifolds

Answer

A measure of the curvature of a manifold

Answer

A measure of the volume of a manifold

Answer

A measure of the genus of a manifold

Question 29

Explain the Poincaré duality theorem.

Accepted Answer

Relates the homology groups of a compact manifold to its cohomology groups

Answer

Relates the fundamental group of a manifold to its homology groups

Answer

States that every compact manifold is homeomorphic to a sphere

Answer

States that every smooth manifold is diffeomorphic to a Euclidean space

Question 30

Which of the following is the fundamental group of the torus?

Accepted Answer

The direct product of two copies of the integers ( ℤ × ℤ )

Answer

The integers ( ℤ )

Answer

The direct product of two copies of the group of order 2 ( ℤ/2 × ℤ/2 )

Answer

The group of order 4 ( ℤ/4 )

Question 31

Which of the following is a necessary and sufficient condition for a manifold to be orientable?

Accepted Answer

The tangent bundle is trivializable.

Answer

The first homology group is torsion-free.

Answer

The fundamental group is abelian.

Answer

The manifold has a nowhere-zero vector field.

Question 32

What does the Seifert-van Kampen theorem describe?

Accepted Answer

The fundamental group of a space obtained by gluing two spaces together.

Answer

The homology of a space obtained by gluing two spaces together.

Answer

The cohomology of a space obtained by gluing two spaces together.

Answer

The homotopy groups of a space obtained by gluing two spaces together.

Question 33

Which of the following is an orientable manifold?

Accepted Answer

Torus

Answer

Mobius strip

Answer

Klein bottle

Answer

Real projective plane

Question 34

Which of the following is a consequence of the Dehn-Sommerville equations?

Accepted Answer

The number of vertices of a simplicial polytope is at least the number of its facets

Answer

The number of edges of a simplicial polytope is equal to the number of its vertices

Answer

The number of faces of a simplicial polytope is equal to the number of its edges

Answer

The number of facets of a simplicial polytope is at most the number of its edges

Question 35

Which of the following is a necessary and sufficient condition for a manifold to be simply connected?

Accepted Answer

Its fundamental group is trivial

Answer

It is contractible

Answer

Its homology groups are all trivial

Answer

Its cohomology groups are all trivial