Master Algebraic Topology: Explore Homology and Cohomology Concepts

Question 1

Which of the following accurately characterizes the cohomology ring?

Accepted Answer

Commutative, associative, and graded

Answer

Commutative and associative only

Answer

Graded only

Answer

Associative and graded only

Question 2

Identify the option that is not a homology theory.

Accepted Answer

Cohomology

Answer

Group homology

Answer

Singular homology

Answer

Simplicial homology

Question 3

Determine the definition of the homology group H_n(X).

Accepted Answer

The group formed by all singular n-cycles in X modulo the group formed by all singular n-boundaries in X.

Answer

The group formed by all continuous maps from X to the n-sphere.

Answer

The group formed by all homotopy classes of continuous maps from the n-sphere to X.

Answer

The group formed by all homology classes in X.

Question 4

Define a cohomology group.

Accepted Answer

The group formed by all singular n-cocycles in X modulo the group formed by all singular n-coboundaries in X.

Answer

The group formed by all continuous maps from X to the n-sphere.

Answer

The group formed by all homotopy classes of continuous maps from the n-sphere to X.

Answer

The group formed by all cohomology classes in X.

Question 5

Define a simplicial complex.

Accepted Answer

A collection of simplices that are glued together along their faces.

Answer

A collection of points that are connected by edges.

Answer

A collection of lines that are connected by points.

Answer

A collection of planes that are connected by lines.

Question 6

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

Accepted Answer

(C_n, d_n)

Answer

(H_n, d_n)

Answer

(C_n, B_n)

Answer

(H_n, B_n)

Question 7

What is the cup product in cohomology theory?

Accepted Answer

A bilinear operation that pairs cohomology classes

Answer

A homomorphism that maps homology to cohomology

Answer

A product that pairs homology classes

Answer

A natural transformation that maps chains to cochains

Question 8

What does the Poincaré duality theorem relate?

Accepted Answer

Homology and cohomology

Answer

Simply connected spaces and their homology

Answer

Homology and homotopy

Answer

The Euler characteristic and the Betti numbers

Question 9

Which of the following is a fundamental group of a topological space X?

Accepted Answer

The group of all continuous maps from X to itself that are homotopic to the identity map

Answer

The group of all continuous maps from the circle to itself

Answer

The group of all continuous maps from the sphere to itself

Answer

The group of all continuous maps from the torus to itself

Question 10

Which of the following is a homology group of a topological space X?

Accepted Answer

The group of all homology classes in X

Answer

The group of all closed loops in X

Answer

The group of all boundaries of cycles in X

Answer

The group of all cohomology classes in X

Question 11

Which of the following is a homology sphere?

Accepted Answer

A sphere

Answer

A torus

Answer

A Klein bottle

Answer

A projective plane

Question 12

Which of the following is a central concept in algebraic topology, used to study the topological properties of spaces?

Accepted Answer

Homology

Answer

Group theory

Answer

Number theory

Answer

Set theory

Question 13

What is the fundamental relationship between homology and cohomology in algebraic topology?

Accepted Answer

They are dual to each other

Answer

They exhibit isomorphism

Answer

They are independent concepts

Answer

They are equivalent concepts

Question 14

What is the homology of a circle?

Accepted Answer

Z

Answer

Z<sub>2</sub>

Answer

R

Answer

0

Question 15

What does the Poincaré duality theorem primarily address in algebraic topology?

Accepted Answer

The relationship between homology and cohomology

Answer

The characterization of simply connected spaces

Answer

The computation of the Euler characteristic

Answer

The existence of homology groups

Question 16

Which fundamental concepts form the basis of algebraic topology?

Accepted Answer

Homology and Cohomology

Answer

Knot theory and Differential geometry

Answer

Group theory and Ring theory

Answer

Category theory and Homotopy theory

Question 17

What is the primary objective of homology in algebraic topology?

Accepted Answer

Characterize the topological features of spaces, including the number and shapes of holes

Answer

Classify topological spaces into distinct categories

Answer

Solve complex differential equations efficiently

Answer

Calculate the volume of manifolds accurately

Question 18

Summarize the Poincaré duality theorem in algebraic topology.

Accepted Answer

It establishes an isomorphism between the homology and cohomology groups of a closed manifold

Answer

It relates the fundamental group of a space to its homology group

Answer

It determines the Euler characteristic of a space based on its Betti number

Answer

It equates the homology of a space with its cohomology

Question 19

How are cohomology and homology related?

Accepted Answer

Cohomology is the dual of homology

Answer

Cohomology extends the concept of homology

Answer

Homology is a special case of cohomology

Answer

Cohomology and homology are unrelated