Covering Maps and Liftings - Munkres Theorem 54.4

In summary: I.e. his statement is: "IF e1 is a point of p^(-01)(b0), then e1 occurs as a lift." He is not assuming any such e1 exists that is different from e0. Indeed if none did exist, surjectivity would be trivial, since the lift of the identity element of the fundamental group would give the unique point e0. Note his proof works also when e1 = e0
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I am reading Munkres book Topology.

Currently, I am studying Section 54: The Fundamental Group of the Circle and need help with a minor point in the proof of Theorem 54.4

Theorem 54.4 and its proof reads as follows:
?temp_hash=6dcc0996c2b1982c64b7700cce279a30.png

In the proof we read:"If [itex]E[/itex] is path connected, then, given [itex]e_1 \in p^{-1}(b_0)[/itex], ... ... "... ... BUT ... ... how do we know there exists an [itex]e_1[/itex] different from [itex]e_0[/itex] in [itex]p^{-1}(b_0)[/itex] ... maybe [itex]e_0[/itex] is the only element in [itex]p^{-1}(b_0)[/itex]?

What, indeed, do we know about the nature of [itex]p^{-1}(b_0)[/itex]?

Hope someone can help ...

Peter
 

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he is trying to prove surjectivity, i.e. that every element of p^(-1)(b0) occurs as a lift. so he starts by taking any point in p^(-1)(b0) and produces a lift that gives it. I.e. his statement is: "IF e1 is a point of p^(-01)(b0), then e1 occurs as a lift." He is not assuming any such e1 exists that is different from e0. Indeed if none did exist, surjectivity would be trivial, since the lift of the identity element of the fundamental group would give the unique point e0. Note his proof works also when e1 = e0.
 
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mathwonk said:
he is trying to prove surjectivity, i.e. that every element of p^(-1)(b0) occurs as a lift. so he starts by taking any point in p^(-1)(b0) and produces a lift that gives it. I.e. his statement is: "IF e1 is a point of p^(-01)(b0), then e1 occurs as a lift." He is not assuming any such e1 exists that is different from e0. Indeed if none did exist, surjectivity would be trivial, since the lift of the identity element of the fundamental group would give the unique point e0. Note his proof works also when e1 = e0.

Thanks for the reply mathwonk ... Most helpful

Peter
 

Related to Covering Maps and Liftings - Munkres Theorem 54.4

What is a covering map?

A covering map is a continuous function between topological spaces that is surjective (onto) and locally homeomorphic (locally looks like a homeomorphism). This means that for every point in the target space, there is a neighborhood around that point that is homeomorphic to a neighborhood around a point in the source space.

What is a lifting in the context of covering maps?

A lifting in the context of covering maps refers to a continuous function that maps points from the target space to points in the source space. This function must preserve the structure of the covering map, meaning that the lifted points must still be mapped onto the same points in the target space.

What is Munkres Theorem 54.4?

Munkres Theorem 54.4 is a theorem in algebraic topology that states that if a covering space is path-connected and locally path-connected, then it is a universal cover of the space it covers. This means that it is the unique simply connected covering space of that space.

How is Munkres Theorem 54.4 used in mathematics?

Munkres Theorem 54.4 is used in mathematics to study the fundamental group of a topological space. By identifying the universal cover of a space, we can gain information about the fundamental group and its structure. This can be useful in many areas of mathematics, including algebraic topology and differential geometry.

What are some applications of covering maps and liftings?

Covering maps and liftings have many applications in mathematics, including in the study of algebraic topology, differential geometry, and complex analysis. They are also used in physics, particularly in the study of gauge theory and string theory. Additionally, covering maps are used in computer graphics and animation to create seamless textures and patterns.

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