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

  1. Jan 6, 2013 #1
    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
     
  2. jcsd
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