Surjectivity of Induced Homomorphism in Algebraic Topology

In summary, the conversation is discussing the surjectivity of the homomorphism r* induced by the continuous map r from X to A, where A is a subset of X and r(a)=a for each a in A. The participants are trying to show that r* is surjective by creating a path homotopy from a loop in A to a loop in X using r. They also mention the properties of r and how it plays a role in the proof. However, they are unsure if it is really that simple and if there is a pasting lemma involved. They eventually come to the conclusion that it is indeed simple and r must fix a in order to have r(f)=f.
  • #1
JasonRox
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I'm totally stuck on these two.

The first is...

Let A be a subset of X; suppose r:X->A is a continuous map from X to A such that r(a)=a for each a e A. If a_0 e A, show that...

r* : Pi_1(X,a_0) -> Pi_1(A,a_0)

...is surjective.

Note: Pi_1 is the first homotopy group and r* is the homomorphism induced by h.

I can visually see in my mind why this is so, but I can't even think of how to write this down at all.

I'm still thinking about it. No need to post anything right now.

The way I'm thinking that is if [f] is in A then I need to show that there is a g in X such that g is path homotopic to f using probably r to create my path homotopy. (f is a loop around a_0 in A)

Once I do that, then it should come out like... r([g]) = [r o g] = [f].

I'm still thinking about this.
 
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  • #2
Can't you use [f] itself for g? I mean, if [f] is a loop in A, then it is a loop in X, and since r fixes A, it should map [f] onto itself.
 
  • #3
DeadWolfe said:
Can't you use [f] itself for g? I mean, if [f] is a loop in A, then it is a loop in X, and since r fixes A, it should map [f] onto itself.

That's exactly what I was thinking too!

But you must show that f is a loop in X by creating the path homotopy that I was speaking about from f to g. Isn't that right?
 
  • #4
f is a loop in X because it is a loop in A, and A is a subset of X.
 
  • #5
DeadWolfe said:
f is a loop in X because it is a loop in A, and A is a subset of X.

So, to show it is surjective, we have...

r([f]) = [r o f] = [f]

If that is so, where does the properties of r even play a role in here?

I feel like there is a pasting lemma in here or something.

Is it really that simple?
 
  • #6
DeadWolfe said:
f is a loop in X because it is a loop in A, and A is a subset of X.

I do understand this.

For some reason, I feel like it's a little too simple.
 
  • #7
Well, if r did not fix a then we would not have r(f)=f.
 
  • #8
DeadWolfe said:
Well, if r did not fix a then we would not have r(f)=f.

That's true.
 

1. What is Algebraic Topology?

Algebraic Topology is a branch of mathematics that combines abstract algebra and topology to study topological spaces and their properties. It focuses on using algebraic techniques to understand and classify the shapes and structures of spaces.

2. What are some applications of Algebraic Topology?

Algebraic Topology has many applications in various fields, including physics, computer science, engineering, and economics. It can be used to analyze data, model complex systems, and solve problems related to networks, circuits, and data compression.

3. What are the main concepts in Algebraic Topology?

The main concepts in Algebraic Topology include homotopy, homology, and cohomology. These concepts help to define and classify spaces based on their properties, such as their connectedness, holes, and dimensions.

4. How is Algebraic Topology different from other branches of topology?

Algebraic Topology differs from other branches of topology, such as geometric topology and differential topology, because it uses algebraic techniques, such as groups, rings, and modules, to study topological spaces. This allows for a more abstract and general approach to understanding spaces.

5. What is the significance of the fundamental group and homology groups in Algebraic Topology?

The fundamental group and homology groups are important tools in Algebraic Topology as they provide a way to classify and distinguish between topological spaces. The fundamental group measures the number of holes in a space, while the homology groups give information about its higher-dimensional structure.

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