A question on degrees of maps of the fundamental group of the unit circle

[trmukerji14]

Hello,

I'm reading a textbook and in the textbook we are discussing the fundamental group of the unit circle and having some difficulty making out what a degree of a map is and why when there is a homotopy between two continuous maps f,g from S[itex]^{1}[/itex] to S[itex]^{1}[/itex] why the deg(f)=deg(g)

We have that the fundamental group of the unit circle, [itex]\pi _{1}[/itex](S[itex]^{1}[/itex] )

f,g: S[itex]^{1}[/itex] →S[itex]^{1}[/itex] are continuous and f≈g(homotopy)
[itex]\iota[/itex] [itex]\in[/itex][itex]\pi _{1}[/itex](S[itex]^{1}[/itex] ) is a generator
The book defines the degree of f, deg(f), as the integer with respect to the composite
[itex]\pi _{1}[/itex](S[itex]^{1},1[/itex] ) [itex]\stackrel{f_{*}}{\rightarrow}[/itex] [itex]\pi _{1}[/itex](S[itex]^{1},f(1)[/itex] )[itex]\stackrel{\gamma_{a}}{\rightarrow}[/itex] [itex]\pi _{1}[/itex](S[itex]^{1},1[/itex] )

Note that a is a path from 1 to f(1) and [itex]\gamma_{a}[/itex] is the change of base-point isomorphism

We have [itex]\iota[/itex]→deg(f)[itex]\iota[/itex]

What exactly is this degree of f?

My understanding is that when we consider [itex]\iota[/itex] the associated integer is 1 and then we have that the associated integer with [itex]\pi _{1}[/itex](S[itex]^{1}, f(1)[/itex] )

What does the change of basepoint homomorphism have to do with it?

Also, why does deg(f)=deg(g)? I know this is due to the abelian property of [itex]\pi _{1}[/itex](S[itex]^{1},1[/itex] ) and the fact that there is a path between f(1) and g(1).

Any help on this would be appreciated greatly.

Thank you

 

[mighty2000]

for your question! The degree of a map is a topological invariant that measures the number of times a map wraps around a given point. In the case of the unit circle, this can be thought of as the number of times a loop goes around the circle. In other words, it measures the winding number of the map.

The reason why deg(f)=deg(g) is because, as you mentioned, the fundamental group of the unit circle is abelian. This means that the order of composition of loops does not affect the resulting loop. So if we have a homotopy between two maps f and g, this means that they can be continuously deformed into each other while keeping the basepoint fixed. This also means that the winding number, or degree, of the map remains the same.

The change of basepoint homomorphism is used to compare the fundamental group at different basepoints. In this case, it is used to compare the fundamental group at the basepoint 1 and at the basepoint f(1). This is necessary because the degree of a map depends on the basepoint chosen.

I hope this helps clarify the concept of degree of a map and why deg(f)=deg(g) in this case. Let me know if you have any further questions.