Topology II: Delve into Homology, Cohomology, and Differential Forms

Question 1

How can the fundamental group be utilized to construct covering spaces? Illustrate your explanation with an example.

Accepted Answer

The fundamental group can be used to construct a covering space by taking the universal cover of the space and quotienting out by the action of the fundamental group.

Answer

The fundamental group can be used to construct a covering space by taking the disjoint union of all open sets in the space.

Answer

The fundamental group can be used to construct a covering space by taking the product of the space with the integers.

Answer

The fundamental group cannot be used to construct covering spaces.

Question 2

What is the Betti number of a topological space?

Accepted Answer

The rank of the n-th homology group

Answer

The number of holes in the space

Answer

The dimension of the space

Answer

The Euler characteristic of the space

Question 3

What is the Poincaré duality theorem?

Accepted Answer

A relationship between homology and cohomology groups on a closed, oriented manifold.

Answer

A relationship between the homology and homotopy groups of a space.

Answer

A relationship between the cohomology groups of a space and its universal cover.

Answer

A relationship between the homology groups of a space and its fundamental group.

Question 4

What is the interpretation of the homology group H_n(X) of a space X?

Accepted Answer

The group representing the number of n-dimensional holes in X

Answer

The group representing the number of n-dimensional spheres in X

Answer

The group representing the number of n-dimensional paths in X

Answer

The group representing the number of n-dimensional cycles in X

Question 5

Which of the following defines a de Rham cohomology group?

Accepted Answer

The cohomology group associated with the de Rham complex of differential forms on a manifold

Answer

The cohomology group associated with the singular complex of a manifold

Answer

The cohomology group associated with the simplicial complex of a manifold

Answer

The cohomology group associated with the chain complex of a manifold

Question 6

Define the Mayer-Vietoris sequence.

Accepted Answer

A long exact sequence that relates the homology groups of a space to the homology groups of its open subsets

Answer

A long exact sequence that relates the cohomology groups of a space to the cohomology groups of its open subsets

Answer

A short exact sequence that relates the homology groups of a space to the homology groups of its open subsets

Answer

A short exact sequence that relates the cohomology groups of a space to the cohomology groups of its open subsets

Question 7

What is the definition of a CW complex?

Accepted Answer

A topological space constructed by attaching cells of various dimensions to each other

Answer

A topological space that is equivalent to a simplicial complex

Answer

A topological space that is contractible to a single point

Answer

A topological space that is simply connected

Question 8

Which of the following is **not** a homology group?

Accepted Answer

H_*^1(X)

Answer

H_0(X)

Answer

H_1(X)

Answer

H_2(X)

Question 9

What is the cohomology ring of a sphere S^n?

Accepted Answer

Z[x]/(x^{n+1})

Answer

Z

Answer

Z/2Z

Answer

Z/nZ

Question 10

Identify the differential form:

Accepted Answer

xdy - ydx

Answer

dx

Answer

x^2

Answer

sin(x)dx^2

Question 11

State the Künneth formula for the homology of the product of two spaces X and Y:

Accepted Answer

H_n(X x Y) = H_n(X) ⊗ H_n(Y)

Answer

H_n(X x Y) = H_n(X) + H_n(Y)

Answer

H_n(X x Y) = H_n(X) ⊕ H_n(Y)

Answer

H_n(X x Y) = H_{n-1}(X) ⊗ H_{n-1}(Y)

Question 12

Determine the homology of a torus:

Accepted Answer

Z ⊕ Z

Answer

Z

Answer

0

Answer

Z/2Z

Question 13

Which of the following is not a property of homology groups?

Accepted Answer

multiplicativity

Answer

additivity

Answer

exactness

Answer

functoriality

Question 14

In the context of a manifold, what constitutes a differential form?

Accepted Answer

A smooth section of the cotangent bundle

Answer

A smooth section of the tangent bundle

Answer

A smooth function defined on the manifold

Answer

A smooth vector field on the manifold

Question 15

What is the de Rham cohomology group associated with a manifold?

Accepted Answer

The group of all closed differential forms on the manifold

Answer

The group consisting of all exact differential forms on the manifold

Answer

The group consisting of all differential forms on the manifold

Answer

The group of all smooth functions on the manifold

Question 16

In topology, what is meant by the fundamental group of a space X?

Accepted Answer

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

Answer

The group consisting of all closed paths within X

Answer

The group consisting of all homeomorphisms from X to itself

Answer

The group consisting of all diffeomorphisms from X to itself

Question 17

Which of the following is correctly identified as a free group?

Accepted Answer

The group consisting of all permutations of a set

Answer

The group consisting of all matrices with a determinant of 1

Answer

The group consisting of all positive integers under addition

Answer

The group consisting of all real numbers under multiplication

Question 18

In topology, what is the homology of a sphere?

Accepted Answer

Z

Answer

Z x Z

Answer

Z x Z x Z

Answer

0

Question 19

What is the essence of the Poincaré duality theorem?

Accepted Answer

It establishes a connection between the homology and cohomology of a space

Answer

It establishes a connection between the fundamental group and homology of a space

Answer

It establishes a connection between the cohomology and de Rham cohomology of a space

Answer

It establishes a connection between the homology and homotopy of a space

Question 20

What is the Euler characteristic of a torus?

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 21

In homotopy theory, how does the spectral sequence associated with a fibration aid in computing the homotopy groups of the total space?

Accepted Answer

The E^2-page of the spectral sequence provides information for computing the homotopy groups of the total space.

Answer

The spectral sequence converges to the homology groups of the fiber and base spaces combined.

Answer

The spectral sequence assists in computing the cohomology groups of the base space.

Answer

The spectral sequence determines if the total space is contractible.

Question 22

A de Rham cohomology class on a smooth manifold is represented by which of the following?

Accepted Answer

A closed differential form

Answer

An exact differential form

Answer

A non-singular vector field

Answer

A smooth function

Question 23

Which sequence is a Mayer-Vietoris sequence?

Accepted Answer

0 -> H_n(X) -> H_n(U) x H_n(V) -> H_n(U cap V) -> 0

Answer

0 -> H_n(U cap V) -> H_n(U) x H_n(V) -> H_n(X) -> 0

Answer

0 -> H_{n+1}(X) -> H_n(U) x H_n(V) -> H_n(U cap V) -> 0

Answer

0 -> H_n(U) x H_n(V) -> H_n(X) -> H_{n+1}(U cap V) -> 0

Question 24

A topological space X is simply connected if and only if which of the following holds true?

Accepted Answer

H_1(X) = 0

Answer

H^1(X) = 0

Answer

H_2(X) = 0

Answer

H^2(X) = 0

Question 25

What does the Hodge decomposition theorem state?

Accepted Answer

Every differential form on a compact Riemannian manifold can be written as a sum of exact, coexact, and harmonic forms.

Answer

Every differential form on a smooth manifold can be written as a sum of exact and coexact forms.

Answer

Every exact differential form on a compact Riemannian manifold is coexact.

Answer

Every coexact differential form on a smooth manifold is exact.

Question 26

In the de Rham cohomology theory, what is the de Rham complex?

Accepted Answer

A cochain complex

Answer

A chain complex

Answer

An exact sequence

Answer

A Mayer-Vietoris sequence

Question 27

What does the Poincaré duality theorem state?

Accepted Answer

For a compact smooth n-manifold, its n-th homology group is isomorphic to its n-th de Rham cohomology group.

Answer

For a compact smooth n-manifold, its n-th cohomology group is isomorphic to its n-th homology group.

Answer

For a compact smooth n-manifold, its n-th de Rham cohomology group is isomorphic to its n-th singular homology group.

Answer

For a compact smooth n-manifold, its n-th homology group is isomorphic to its n-th Čech cohomology group.

Question 28

Which of the following is NOT an application of homology theory?

Accepted Answer

Solving systems of linear equations

Answer

Classifying surfaces based on their topological properties

Answer

Calculating the homology groups of a space

Answer

Computing the Euler characteristic of a space

Question 29

Which of the following is a homology sphere?

Accepted Answer

The sphere

Answer

The torus

Answer

The real projective plane

Answer

The Klein bottle

Question 30

The Euler characteristic of a torus is:

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 31

Which of the following is NOT a homology group?

Accepted Answer

H(X)

Answer

H_0(X)

Answer

H_1(X)

Answer

H_2(X)

Question 32

The cohomology ring of a point is:

Accepted Answer

Z

Answer

Z_2

Answer

R

Answer

C

Question 33

Determine the dimension of the tangent space at any given point on a differentiable manifold.

Accepted Answer

Equivalent to the manifold's dimension

Answer

Equivalent to the manifold's dimension minus one

Answer

Equivalent to the number of connected components in the manifold

Answer

Equivalent to the number of charts in the manifold

Question 34

How can cohomology be applied to classify algebraic varieties?

Accepted Answer

Classify algebraic varieties based on their cohomology rings.

Answer

Compute the Betti numbers of algebraic varieties.

Answer

Determine the dimension of algebraic varieties.

Answer

Study the singularities of algebraic varieties.

Question 35

Calculate the Euler characteristic of a torus.

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 36

Identify the option that is not a de Rham cohomology group:

Accepted Answer

H^3(M, R^2)

Answer

H^0(M, R)

Answer

H^1(M, R)

Answer

H^2(M, R)

Question 37

What is the homology group $H_1(S^1)$ of the circle?

Accepted Answer

$\mathbb{Z}$

Answer

$\mathbb{Z}_2$

Answer

$\mathbb{Q}$

Answer

$\mathbb{R}$

Question 38

What is the cohomology ring of a topological space X with coefficients in $\mathbb{Z}$?

Accepted Answer

$H^*(X; \mathbb{Z})$

Answer

$H(X; \mathbb{Z})$

Answer

$C^*(X; \mathbb{Z})$

Answer

$C(X; \mathbb{Z})$

Question 39

What is the definition of the de Rham cohomology of a Riemann manifold?

Accepted Answer

The cohomology group of the space of smooth differential forms on the manifold

Answer

The homology group of the space of continuous differential forms on the manifold

Answer

The cohomology group of the space of continuous differential forms on the manifold

Answer

The homology group of the space of smooth differential forms on the manifold

Question 40

Which of the following is the definition of a spectral sequence?

Accepted Answer

A sequence of chain complexes

Answer

A sequence of homology groups

Answer

A sequence of cohomology groups

Answer

A sequence of homotopy groups

Question 41

What is the de Rham cohomology of a smooth manifold?

Accepted Answer

The cohomology group associated with the space of differential forms on the manifold

Answer

The homology group associated with the space of smooth functions on the manifold

Answer

The fundamental group of the manifold

Answer

The Betti numbers of the manifold

Question 42

What is the cohomology group of a torus?

Accepted Answer

Z x Z

Answer

Z/2Z x Z/2Z

Answer

Z

Answer

0

Question 43

What is the relationship between homology and cohomology?

Accepted Answer

They are dual to each other

Answer

They are isomorphic to each other

Answer

They are independent of each other

Answer

They are complementary to each other

Question 44

What is the significance of the Euler characteristic?

Accepted Answer

It is a topological invariant that measures the number of holes in a space

Answer

It is a topological invariant that measures the number of connected components in a space

Answer

It is a topological invariant that measures the orientability of a space

Answer

It is a topological invariant that measures the homology groups of a space

Question 45

What is the relationship between homology and cohomology?

Accepted Answer

Homology is a dual theory to cohomology.

Answer

Cohomology is a generalization of homology.

Answer

Homology and cohomology are unrelated theories.

Answer

Homology is a special case of cohomology.

Question 46

What is the de Rham cohomology group?

Accepted Answer

The group of closed differential forms.

Answer

The group of exact differential forms.

Answer

The group of all differential forms.

Answer

The group of all smooth functions.

Question 47

What is the homology of a circle?

Accepted Answer

Z

Answer

Z x Z

Answer

Z^2

Answer

0

Question 48

What is the Poincaré duality theorem?

Accepted Answer

A theorem that establishes a relationship between the homology and cohomology groups of a compact, orientable manifold, stating that the homology group in dimension n is isomorphic to the cohomology group in dimension m-n, where m is the dimension of the manifold.

Answer

A theorem that relates homology and homotopy.

Answer

A theorem that relates cohomology and homotopy.

Answer

A theorem that relates homology and differential forms.

Question 49

What is the homology of a circle?

Accepted Answer

Z

Answer

Z^2

Answer

0

Answer

Z/2

Question 50

What is the Künneth theorem?

Accepted Answer

A theorem that relates the homology of a product space to the homology of its factors, stating that the homology of the product is isomorphic to the tensor product of the homology of the factors.

Answer

A theorem that relates the cohomology of a product space to the cohomology of its factors.

Answer

A theorem that relates the homology of a sphere to its dimension.

Answer

A theorem that relates the cohomology of a torus to its genus.

Question 51

What is the cohomology ring of a circle?

Accepted Answer

Z/2[x]

Answer

Z[x]

Answer

Q[x]

Answer

R[x]

Question 52

In differential geometry, what mathematical object measures the intrinsic curvature of a complex vector bundle?

Accepted Answer

Chern class

Answer

Euler characteristic

Answer

Fundamental class

Answer

Characteristic class

Question 53

For the circle denoted by S^1, which of the following represents its fundamental group?

Accepted Answer

Z

Answer

R

Answer

Z/2

Answer

1

Question 54

In differential geometry, what type of differential form is a closed differential form on a smooth manifold M known as?

Accepted Answer

De Rham cohomology class

Answer

Exact differential form

Answer

Harmonic differential form

Answer

Any differential form on M

Question 55

What is the dimension of the de Rham cohomology group of an n-dimensional manifold?

Accepted Answer

n

Answer

n+1

Answer

n-1

Answer

2n

Question 56

What is the Euler characteristic of a torus?

Accepted Answer

0

Answer

1

Answer

2

Answer

-1

Question 57

Which of the following is a homology group associated with a simplicial complex?

Accepted Answer

Simplicial homology group

Answer

Singular homology group

Answer

De Rham cohomology group

Answer

Čech cohomology group

Question 58

State the Poincaré duality theorem in dimension n.

Accepted Answer

H^n(X) is isomorphic to H_n(X)

Answer

H^n(X) is isomorphic to H_{n-1}(X)

Answer

H_n(X) is isomorphic to H^{n-1}(X)

Answer

H^n(X) is isomorphic to H_{n+1}(X)

Question 59

What does the Künneth formula for homology imply?

Accepted Answer

The homology of a product space is the tensor product of the homologies of the factors

Answer

The homology of a quotient space is the quotient of the homologies of the covering space

Answer

The homology of a suspension is the suspension of the homology of the base

Answer

The homology of a wedge sum is the direct sum of the homologies of the factors

Question 60

Describe the Mayer-Vietoris sequence for the homology of an open cover of a topological space.

Accepted Answer

A long exact sequence connecting the homology of the open sets and their intersection

Answer

A short exact sequence splitting the homology of the union into the homology of the open sets

Answer

A filtration of the homology of the union by the homology of the open sets

Answer

An isomorphism between the homology of the union and the direct sum of the homology of the open sets

Question 61

Identify a fundamental theorem in topology.

Accepted Answer

Poincaré Duality Theorem

Answer

Gauss-Bonnet Theorem

Answer

Stokes' Theorem

Answer

Hodge Decomposition Theorem

Question 62

Which of the following is a de Rham cohomology group?

Accepted Answer

H^n(M; R)

Answer

H_n(M; Z)

Answer

Ext^n(M; Z)

Answer

Tor_n(M; Z)

Question 63

What is the result of the cup product of two cohomology classes?

Accepted Answer

Another cohomology class

Answer

A homology class

Answer

An integer

Answer

A vector

Question 64

Poincaré Duality Theorem: Which of the following statements best describes the Poincaré Duality Theorem?

Accepted Answer

The homology of a compact n-manifold is isomorphic to its cohomology in dimension n.

Answer

The fundamental group of a simply connected space is trivial.

Answer

Every compact Riemannian manifold has a metric of constant curvature.

Answer

A closed surface has Euler characteristic zero.