Singular Homology

Test your knowledge of singular homology groups for various topological spaces including surfaces and their generalizations with handles and twists.

15 Questions Published

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Questions

Question 1 Multiple Choice (Single Answer)

What is the definition of a singular homology group?

The group of all continuous maps from a simplicial complex to the integers.
The group of all continuous maps from a simplicial complex to the circle.
The group of all continuous maps from a simplicial complex to the sphere.
The group of all continuous maps from a simplicial complex to the torus.

Question 2 Multiple Choice (Single Answer)

What is the relationship between singular homology and simplicial homology?

Singular homology is a generalization of simplicial homology.
Simplicial homology is a generalization of singular homology.
Singular homology and simplicial homology are equivalent.
Singular homology and simplicial homology are unrelated.

Question 3 Multiple Choice (Single Answer)

What is the homology group of a sphere?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 4 Multiple Choice (Single Answer)

What is the homology group of a torus?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 5 Multiple Choice (Single Answer)

What is the homology group of a Klein bottle?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 6 Multiple Choice (Single Answer)

What is the homology group of a projective plane?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 7 Multiple Choice (Single Answer)

What is the homology group of a Mobius strip?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 8 Multiple Choice (Single Answer)

What is the homology group of a disk?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 9 Multiple Choice (Single Answer)

What is the homology group of a cylinder?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 10 Multiple Choice (Single Answer)

What is the homology group of a cone?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^2$
$\mathbb{Z}_2^2$

Question 11 Multiple Choice (Single Answer)

What is the homology group of a sphere with $n$ handles?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^n$
$\mathbb{Z}_2^n$

Question 12 Multiple Choice (Single Answer)

What is the homology group of a torus with $n$ holes?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^n$
$\mathbb{Z}_2^n$

Question 13 Multiple Choice (Single Answer)

What is the homology group of a Klein bottle with $n$ handles?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^n$
$\mathbb{Z}_2^n$

Question 14 Multiple Choice (Single Answer)

What is the homology group of a projective plane with $n$ holes?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^n$
$\mathbb{Z}_2^n$

Question 15 Multiple Choice (Single Answer)

What is the homology group of a Mobius strip with $n$ twists?

$\mathbb{Z}$
$\mathbb{Z}_2$
$\mathbb{Z}^n$
$\mathbb{Z}_2^n$

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