Algebraic Topology - Retractions and Homomorpisms Induced by Inclusions

This implies that the induced homomorphism on fundamental groups of A and X is also the identity, making it injective. In summary, the Lemma 55.1 states that if A is a retract of X, then the homomorphism of fundamental groups induced by inclusion is injective. The proof of this lemma involves showing that the induced homomorphism is the identity, which can be demonstrated through diagrams and a formal proof.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Munkres book on Topology, Part II - Algegraic Topology Chapter 9 on the Fundamental Group.

On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions":

" Lemma 55.1. If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion [itex] j: A \rightarrow X [/itex] is injective"


I am struggling with the proof - not so much intuitively - but in formulating a formal and explicit proof.


Because explaining my postion requires diagrams I have set out my problem in an attachment - see the attachment "Retractions and Induced Homomorphisms.


I have also provided an attachement of the relevant pages of Munkres book

I would like as much as anything a confirmation that my reasoning in the attachment "Problem ... ... " is correct. I would also be most interested to see how to formulate a formal and explicit proof of the Lemma

Peter
 

Attachments

  • Problem - Retractions and Induced Homomorphisms.pdf
    141.2 KB · Views: 266
  • Munkres Pages 333 and 348 - Fundamental Group.pdf
    84.8 KB · Views: 359
Physics news on Phys.org
  • #2
The inclusion of A in X followed by the retract map is the identity on A
 

Related to Algebraic Topology - Retractions and Homomorpisms Induced by Inclusions

1. What is algebraic topology?

Algebraic topology is a branch of mathematics that studies the properties of geometric objects and spaces by using algebraic tools. It focuses on the topological properties of spaces, such as their connectedness and continuity, and uses algebraic structures like groups and rings to analyze these properties.

2. What are retractions in algebraic topology?

In algebraic topology, a retraction is a continuous map from a space to a subspace that fixes all the points in the subspace. This means that the subspace is essentially a part of the larger space and can be continuously deformed onto itself without leaving the larger space.

3. How are homomorphisms induced by inclusions in algebraic topology?

In algebraic topology, an inclusion is a map from a subspace to the larger space that assigns each point in the subspace to itself in the larger space. This induces a corresponding homomorphism between the fundamental groups of the two spaces, which captures the topological properties of the inclusion.

4. How are retractions and homomorphisms related in algebraic topology?

Retractions and homomorphisms are closely related in algebraic topology. In fact, a retraction can be thought of as a special case of a homomorphism induced by an inclusion, where the subspace is also the image of the larger space under the homomorphism. This relationship allows us to study the topological properties of spaces by looking at their subspaces and homomorphisms between them.

5. What are some applications of algebraic topology?

Algebraic topology has many applications in various fields, including physics, engineering, and computer science. It can be used to study the shapes and properties of objects, design efficient networks and algorithms, and analyze data and patterns. It also has connections to other areas of mathematics, such as differential geometry and abstract algebra, making it a powerful tool for understanding and solving complex problems.

Similar threads

Algebraic Topology - Fundamental Group and the Homomorphism induced by h
    • Differential Geometry
Replies
4
Views
3K
A Sequence induced by short exact sequence
    • Topology and Analysis
Replies
3
Views
1K
Fixed Point Theorem: Unit Square Injection to Bigger Square
    • Differential Geometry
Replies
12
Views
3K
I New Video Course on General Topology on YouTube
    • Topology and Analysis
Replies
2
Views
418
How accessible is Bott & Tu's book on algebraic topology?
    • Differential Geometry
Replies
5
Views
5K
Algebraic Topology: Connected Sum & Reference Help
    • Differential Geometry
Replies
5
Views
3K
Knot Theory & Topology w/ Munkres & "The Knot Book" by Adams
    • Topology and Analysis
Replies
11
Views
3K
I Intuition behind induced homomorphism from covering maps
    • Topology and Analysis
Replies
15
Views
2K
B Algebraic Topology in the tv show The Big Bang Theory
    • Cosmology
Replies
11
Views
20K
Courses Which Abstract Algebra Sequence is Best for Aspiring High Energy Physicists?
    • STEM Academic Advising
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top