1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jan 6, 2013 #1

    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
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted