Algebraic Topology: Master the Fundamentals of Topological Spaces

Question 1

Which of these is NOT a core concept in algebraic topology?

Accepted Answer

Knot Theory

Answer

Homology

Answer

Cohomology

Answer

Fundamental Group

Question 2

Which concept is NOT part of algebraic topology?

Accepted Answer

Differential Geometry

Answer

Homology

Answer

Cohomology

Answer

Fundamental Group

Question 3

Identify the space that cannot be continuously deformed to a point.

Accepted Answer

Möbius strip

Answer

Sphere

Answer

Disk

Answer

Torus

Question 4

Determine the homology group associated with a circle.

Accepted Answer

ℤ

Answer

ℤ²

Answer

ℤ³

Answer

ℤ₄

Question 5

Which theorem is not a cornerstone of algebraic topology?

Accepted Answer

Cauchy's integral theorem

Answer

Poincaré duality

Answer

Alexander duality

Answer

Thom's theorem

Question 6

Which technique in algebraic topology allows us to categorize and classify topological spaces based on their homology groups?

Accepted Answer

Homology theory

Answer

Cohomology

Answer

Homotopy theory

Answer

Knot theory

Question 7

Which of the following is a fundamental concept of algebraic topology?

Accepted Answer

Homology

Answer

Differential calculus

Answer

Trigonometry

Answer

Linear algebra

Question 8

What is the primary purpose of cohomology in algebraic topology?

Accepted Answer

To study the topological properties of spaces

Answer

To solve differential equations

Answer

To calculate volumes of objects

Answer

To find derivatives of functions

Question 9

Which of the following is NOT a characteristic of the fundamental group?

Accepted Answer

It is always isomorphic to the homology group

Answer

It is a topological invariant

Answer

It is a group

Answer

It is generated by the loops in a space

Question 10

Which of the following theorems relates homology and cohomology?

Accepted Answer

Poincaré duality theorem

Answer

Fundamental theorem of calculus

Answer

Pythagorean theorem

Answer

Euler's theorem

Question 11

Which of the following is a significant application of algebraic topology?

Accepted Answer

Studying the topology of manifolds

Answer

Solving partial differential equations

Answer

Finding critical points of functions

Answer

Calculating surface areas

Question 12

What is the simplicial complex associated with the following set of simplices: {0}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}?

Accepted Answer

A tetrahedron

Answer

A triangle

Answer

A cube

Answer

A sphere

Question 13

What is the dimension of the cohomology group H^n(X)?

Accepted Answer

n

Answer

n-1

Answer

n+1

Answer

n-2

Question 14

What is the Künneth formula for the homology of a product space X x Y?

Accepted Answer

H_*(X x Y) = H_*(X) tensor product H_*(Y)

Answer

H_*(X x Y) = H_*(X) direct sum H_*(Y)

Answer

H_*(X x Y) = H_*(X) tensor product H^*(Y)

Answer

H_*(X x Y) = H^*(X) tensor product H^*(Y)

Question 15

What is the homology of a simplicial complex?

Accepted Answer

A homology group

Answer

A free abelian group

Answer

A chain complex

Answer

A cohomology group

Question 16

Which of the following is NOT a concept introduced in Algebraic Topology?

Accepted Answer

Cosmology

Answer

Homology

Answer

Cohomology

Answer

Fundamental group

Question 17

How are homology and cohomology related?

Accepted Answer

Cohomology is the dual of homology

Answer

Homology is the dual of cohomology

Answer

They are unrelated concepts

Answer

They are equivalent concepts

Question 18

What is the name for the simplicial complex connected to a space X?

Accepted Answer

The singular complex of X

Answer

The chain complex of X

Answer

The homology complex of X

Answer

The cohomology complex of X

Question 19

Which property is characteristic of homology groups?

Accepted Answer

Their invariance under continuous deformations

Answer

Their derivation from the fundamental group

Answer

Their isomorphism with cohomology groups

Answer

Their triviality for all spaces

Question 20

Which of the following is an example of a chain complex?

Accepted Answer

The sequence of singular homology groups of a space

Answer

The sequence of cohomology groups of a space

Answer

The sequence of fundamental groups of a space

Answer

The sequence of Euler characteristics of a space

Question 21

Which theorem explores the relationship between homology and homotopy?

Accepted Answer

Hurewicz theorem

Answer

Poincaré duality theorem

Answer

Lefschetz fixed point theorem

Answer

Dehn-Somerville theorem

Question 22

Identify an example of a homology group.

Accepted Answer

Singular homology group

Answer

Fundamental group

Answer

de Rham cohomology group

Answer

Betti number

Question 23

Describe the relationship between homology and cohomology.

Accepted Answer

Homology and cohomology are dual to each other.

Answer

Homology and cohomology are isomorphic to each other.

Answer

Homology and cohomology are unrelated.

Answer

Homology is a special case of cohomology.

Question 24

Define the fundamental group in algebraic topology.

Accepted Answer

Group of all continuous loops based at a given point in a topological space.

Answer

Group of all continuous paths based at a given point in a topological space.

Answer

Group of all homeomorphisms of a topological space.

Answer

Group of all diffeomorphisms of a topological space.

Question 25

The homology group H_k(X) of a space X is defined as:

Accepted Answer

The group of k-cycles of X modulo the group of k-boundaries of X

Answer

The group of k-boundaries of X modulo the group of k-cycles of X

Answer

The set of all k-dimensional submanifolds of X

Answer

The set of all continuous maps from X to an n-sphere

Question 26

The fundamental group of a space X is:

Accepted Answer

The group of all homotopy classes of loops based at a fixed point in X

Answer

The group of all loops based at a fixed point in X

Answer

The group of all paths based at a fixed point in X

Answer

The group of all continuous maps from X to the circle

Question 27

Which of the following is the Künneth theorem?

Accepted Answer

H^m+n(X × Y) ≅ H^m(X) ⊗ Hⁿ(Y)

Answer

H_m+n(X × Y) ≅ H_m(X) ⊗ H_n(Y)

Answer

H^m-n(X × Y) ≅ H^m(X) ⊗ Hⁿ(Y)

Answer

H_m-n(X × Y) ≅ H_m(X) ⊗ H_n(Y)

Question 28

In algebraic topology, which concept is central to analyzing the topological features of spaces?

Accepted Answer

Homology

Answer

Number theory

Answer

Group theory

Answer

Calculus

Question 29

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

Accepted Answer

Knot theory

Answer

Homology

Answer

Cohomology

Answer

The fundamental group

Question 30

How do you construct a simplicial complex from a set of simplices?

Accepted Answer

Identify simplices that connect to form a simplicial complex.

Answer

Calculate the homology groups of the complex.

Answer

Check if the complex is homeomorphic to a manifold.

Answer

Decompose the complex into smaller simplices using triangulation.

Question 31

What is the fundamental group of the torus?

Accepted Answer

Z x Z

Answer

0

Answer

S^1 x S^1

Answer

Z

Question 32

What is the homology group of a sphere?

Accepted Answer

Z

Answer

Z_2

Answer

R

Answer

0

Question 33

Which of the following is a CW complex?

Accepted Answer

A space built by attaching n-cells to an (n-1)-sphere, for all n>=0

Answer

A space that is homeomorphic to a sphere

Answer

A space that is simply connected

Question 34

Which of the following is a spectral sequence?

Accepted Answer

A sequence of chain complexes and homomorphisms

Answer

A sequence of groups and homomorphisms

Answer

A sequence of sets and functions

Answer

A sequence of vector spaces and linear maps

Question 35

What is the cup product in cohomology?

Accepted Answer

A bilinear operation that takes two cohomology classes and produces a cohomology class

Answer

A linear operation that takes a cohomology class and produces a cohomology class

Answer

An operation that takes a homology class and produces a cohomology class

Answer

An operation that takes two cohomology classes and produces a homology class

Question 36

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

Accepted Answer

Knot theory

Answer

Cohomology

Answer

Homology

Answer

Fundamental group

Question 37

Which of the following accurately describes the purpose of homology in algebraic topology?

Accepted Answer

To analyze topological properties of spaces by studying their homology groups

Answer

To calculate the cohomology ring of a space

Answer

To define the fundamental group of a space

Question 38

Which of the following statements about the fundamental group of a space is correct?

Accepted Answer

It serves as a group that captures the connectivity and loop structure of the space

Answer

It is a cohomology group used to study the cohomology ring of a space

Answer

It is a homology group used to study the homology of a space

Question 39

What is the nature of the relationship between cohomology and homology?

Accepted Answer

Cohomology is the dual of homology, and the two are connected by a duality theorem

Answer

Cohomology is used to study the homology groups of a space

Answer

Cohomology is a generalization of homology to higher dimensions

Question 40

What is the primary purpose of the Mayer-Vietoris sequence in algebraic topology?

Accepted Answer

To compute the homology of a space by breaking it down into smaller components

Answer

To calculate the cohomology ring of a space using a filtration

Answer

To define the fundamental group of a space by combining smaller fundamental groups

Question 41

What is the significance of the Eilenberg-Steenrod axioms in algebraic topology?

Accepted Answer

They establish a set of properties that characterize homology groups

Answer

They define the fundamental group of a space in terms of generators and relations

Answer

They determine the cohomology ring of a space using a spectral sequence

Question 42

What is the primary goal of studying algebraic topology?

Accepted Answer

To gain an understanding of the topological properties of spaces through the application of algebraic techniques

Answer

To classify all types of knots

Answer

To develop new methods for solving differential equations