Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics & Statistics

Algebraic Topology I Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art representing the Algebraic Topology I course

Boost your understanding of Algebraic Topology I with our engaging practice quiz, designed to test your mastery of topological spaces using algebraic invariants. This quiz covers key themes such as the fundamental group, covering spaces, simplicial and singular homology, and essential applications like the Brouwer fixed point theorem and the Jordan curve theorem, making it an indispensable resource for students aiming to excel in algebraic topology.

What is the fundamental group of the circle S¹?
Trivial group
ℤ (the group of integers)
ℤ/2ℤ
ℝ (the group of real numbers under addition)
The fundamental group of S¹ is isomorphic to ℤ because every loop on the circle can be classified by an integer counting how many times it winds around the circle. This integer (winding number) captures the essential topological feature of S¹.
Which description best defines a covering space?
A continuous surjective map that locally looks like a disjoint union of open subsets of the base space
Any continuous map between two topological spaces
A map that is globally a homeomorphism
A quotient map defined by an equivalence relation
A covering space is defined via a map where every point in the base has an open neighborhood evenly covered by the map. This local condition ensures that the space above the base is made up of disjoint copies of these neighborhoods.
What are the building blocks of a simplicial complex?
Simplices such as points, line segments, triangles, etc., which are assembled in a combinatorially consistent way
Open sets that cover the space
Circles and spheres only
Continuous images of Euclidean spaces
A simplicial complex is constructed from simplices (vertices, edges, triangles, etc.) that are attached along their faces under strict rules. This combinatorial structure is the foundation for developing simplicial homology.
Which theorem guarantees that any continuous function from a closed disk to itself has a fixed point?
Brouwer Fixed Point Theorem
Jordan Curve Theorem
Banach Fixed Point Theorem
Poincaré Duality
The Brouwer Fixed Point Theorem asserts that any continuous function mapping a closed disk to itself must have at least one fixed point. This result is fundamental in topology and has many significant applications.
Which theorem states that a simple closed curve in the plane divides the plane into two distinct regions?
Jordan Curve Theorem
Brouwer Fixed Point Theorem
Sperner's Lemma
Van Kampen's Theorem
The Jordan Curve Theorem guarantees that any simple closed curve in the plane splits the plane into an inside and an outside region. This classical result is essential for understanding planar topology and its subdivisions.
If a 2-cell is attached to a circle via a map of degree n, what is the fundamental group of the resulting space?
ℤ/nℤ
ℤ
The trivial group
ℤ × ℤ
Attaching a 2-cell to a circle via a map of degree n effectively kills the nth power of the fundamental group's generator. This construction results in a fundamental group isomorphic to ℤ/nℤ, reflecting the degree of attachment.
Which property is necessary for a map p: E → B to be a covering map?
It must be surjective and every point in B must have an evenly covered neighborhood
It must be an injective continuous map
It is defined by a one-to-one correspondence between points in E and B
It must be an open embedding
A covering map is defined by having every point in the base space possess an evenly covered neighborhood, in addition to being surjective. This condition guarantees the local homeomorphism property essential for a covering space.
How does the excision theorem assist in the computation of homology groups?
It permits the removal of a subspace that does not affect the overall homology
It replaces the space with a homotopy equivalent, yet simpler space
It guarantees that all homology groups become trivial
It shifts the focus from homology to homotopy groups
The excision theorem allows one to remove a subspace (under appropriate conditions) without changing the homology groups of the space. This simplifies calculations by reducing the complexity of the space while retaining its homological characteristics.
Which chain complex is central to defining singular homology?
The free abelian groups generated by singular simplices with the standard boundary operator
The chain complex generated by cells in a CW complex
Exact sequences formed by homotopy groups
Čech cohomology chain complexes
Singular homology is defined using a chain complex where the chains are free abelian groups generated by singular simplices. The boundary operator on these chains is used to build homology groups that reflect the topology of the space.
What is the fundamental group of the Möbius band?
ℤ
The trivial group
ℤ/2ℤ
ℤ × ℤ
The Möbius band deformation retracts onto its central circle, which means its essential loop structure is the same as that of a circle. Consequently, its fundamental group is isomorphic to ℤ.
What does it mean for a topological space to be simply connected?
The space is path-connected and has a trivial fundamental group
The space is disconnected with trivial homology
Every loop can be contracted to a point, even if the space is not path-connected
The space has non-trivial higher homotopy groups
A simply connected space must be path-connected and have no 'holes', meaning its fundamental group is trivial. This ensures that every loop can be continuously contracted to a point without obstruction.
In classifying covering spaces, which algebraic invariant is essential?
The fundamental group of the base space
The Euler characteristic
Singular homology groups
Cohomology rings
The classification of covering spaces relies heavily on the fundamental group of the base space, as covers correspond to subgroups of this group (up to conjugacy). This connection is a cornerstone in algebraic topology for understanding how covers relate to the underlying space.
Which homology theory uses combinatorial structures of simplices to compute homological invariants?
Simplicial homology
Singular homology
de Rham homology
Čech cohomology
Simplicial homology is based on the combinatorial structure of simplicial complexes, using simplices as the basic building blocks to construct chain complexes. This method contrasts with singular homology, which uses arbitrary continuous maps from standard simplices.
If a space X is contractible, what are its singular homology groups?
H₀ is isomorphic to ℤ and all higher homology groups are trivial
All homology groups are trivial
H₝ is isomorphic to ℤ while the rest are trivial
Both H₀ and H₝ are isomorphic to ℤ, with higher groups trivial
A contractible space is homotopy equivalent to a point, so its singular homology mirrors that of a point. This means H₀ is ℤ (reflecting a single connected component) and all higher homology groups vanish.
In applying the Brouwer Fixed Point Theorem to disks, which property of the disk is crucial?
Being compact and convex
Being simply connected
Having a non-trivial fundamental group
Possessing a boundary homeomorphic to a circle
The proof of the Brouwer Fixed Point Theorem relies on the disk being both compact and convex, which are essential for ensuring that any continuous self-map has a fixed point. These geometric properties restrict the behavior of continuous functions sufficiently to guarantee a fixed point.
0
{"name":"What is the fundamental group of the circle S¹?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"What is the fundamental group of the circle S¹?, Which description best defines a covering space?, What are the building blocks of a simplicial complex?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Analyze the fundamental group to determine topological properties of spaces.
  2. Apply covering space classification techniques to solve topology problems.
  3. Evaluate simplicial and singular homology in the context of space invariants.
  4. Interpret classical theorems like the Brouwer fixed point and Jordan curve theorems to justify topological conclusions.

Algebraic Topology I Additional Reading

Embarking on the journey of Algebraic Topology? Here are some top-notch resources to guide you through the twists and turns of this fascinating subject:

  1. MIT OpenCourseWare: Algebraic Topology I Dive into lecture notes and problem sets from MIT's graduate-level course, covering topics like singular homology, CW complexes, and Poincaré duality.
  2. The Fundamental Group and Covering Spaces Jesper M. Møller's lecture notes provide a deep dive into the fundamental group and the classification of covering spaces, essential concepts in Algebraic Topology.
  3. Algebraic Topology Course by John Baez Explore comprehensive lecture notes from John Baez's course, focusing on the fundamental group, covering spaces, and homotopy theory.
  4. MIT OpenCourseWare: Algebraic Topology II For those looking to advance further, this course delves into homotopy theory, fiber bundles, and spectral sequences, building upon foundational concepts.
  5. Columbia University: Algebraic Topology I Access course materials, including homework assignments and additional resources, from Columbia University's graduate course on Algebraic Topology.
Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Mathematics & Statistics Quizzes

Powered by: Quiz Maker