Understanding the Role of the Identity Map in Fundamental Group Theory

In summary, (idX)* : π1(X) → π1(X) is the identity map on the fundamental group of X, which means it preserves the group structure and takes each loop to itself.
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Mikaelochi
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Another problem from a topology course I took and never really got
HW9Q4.png

So, this problem I sort of get conceptually but I don't know how I can possibly rewrite (idX)∗ : π1(X) → π1(X). Does this involve group theory? It's supposed to be simple but I honestly I don't see how. Again, any help is greatly appreciated. Thanks.
 
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If [itex]f : X \to Y[/itex] is continuous, then [itex]f_{*} : \pi_1(X) \to \pi_1(Y)[/itex] is defined by [tex]f_{*}(\gamma) : [0,1] \to Y : t \mapsto (f \circ \gamma)(t)[/tex] for each [itex]\gamma \in \pi_1(X)[/itex].

If [itex]X = Y[/itex] and [itex]f[/itex] is the identity on [itex]X[/itex], then [itex]f_{*}(\gamma) = \gamma[/itex] and [itex]f_{*}[/itex] is the identity on [itex]\pi_1(X)[/itex].
 
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Hi there! I can understand your confusion with rewriting (idX)* : π1(X) → π1(X). It might help to think of it in terms of group theory. Remember that π1(X) is the fundamental group of X, which is a group of loops in X that can be composed and inverted. So (idX)* is simply the identity map on the fundamental group, which just takes each loop to itself. This may seem trivial, but it becomes important when studying homotopy and homotopy equivalence.

Now, to rewrite this, you can think of it as (idX)* : G → G, where G is the fundamental group of X. This is a homomorphism from G to itself, which means it preserves the group structure. In other words, it takes the composition and inversion of loops in G to the composition and inversion of loops in G. So, in a way, it is just a fancy way of saying that (idX)* is the identity map on the fundamental group.

I hope this helps! Let me know if you have any other questions.
 

1. What is an identity map?

An identity map is a function that maps each element of a set to itself. In other words, it is a function that preserves the identity of each element in a set.

2. How is an identity map written mathematically?

An identity map is typically denoted by the symbol "id", and is written as id(x) = x, where x is an element in the set.

3. What is the purpose of using identity maps in proofs?

Identity maps are often used in proofs to show that a given function is a bijection, or a one-to-one and onto mapping. This is because identity maps preserve the identity of each element, making it easier to prove that the function is a bijection.

4. Can an identity map be composed with other functions?

Yes, an identity map can be composed with other functions. This means that the output of the identity map is used as the input for another function. The result is still the same input, as the identity map preserves the identity of each element.

5. Are identity maps only used in mathematics?

No, identity maps can also be used in other fields, such as computer science and data analysis. In computer science, identity maps are used to represent the identity of an object or data structure. In data analysis, identity maps are used to compare data sets and identify patterns or trends.

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