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Algebraic Topology - Fundamental Group and the Homomorphism induced by h

  1. Jul 6, 2012 #1
    On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334)

    "Suppose that [itex] h: X \rightarrow Y [/itex] is a continuous map that carries the point [itex] x_0 [/itex] of X to the point [itex] y_0 [/itex] of Y.

    We denote this fact by writing:

    [itex] h: ( X, x_0) \rightarrow (Y, y_0) [/itex]

    If f is a loop in X based at [itex] x_0 [/itex] , then the composite [itex] h \circ f : I \rightarrow Y [/itex] is a loop in Y based at [itex] y_0 [/itex]"

    I am confused as to how this works ... can someone help with the formal mechanics of this.

    To illustrate my confusion, consider the following ( see my diagram and text in atttachment "Diagram ..." )


    Consider a point [itex] i^' [/itex] [itex] \in [0, 1][/itex] that is mapped by f into [itex] x^' [/itex] i.e. [itex] f( i^{'} ) [/itex] [itex] = x^' [/itex]

    Then we would imagine that [itex] i^' [/itex] is mapped by [itex] h \circ f [/itex] into some corresponding point [itex] y^' [/itex] ( see my diagram and text in atttachment "Diagram ..." )


    i.e. [itex] h \circ f (i^{'} ) [/itex] [itex] = y^' [/itex]

    BUT

    [itex] h \circ f (i^{'} ) = h(f(i^{'} )) = h(x^{'} ) [/itex]

    But (see above) we only know of h that it maps [itex] x_0 [/itex] into [itex] y_0 [/itex]? {seems to me that is not all we need to know about h???}

    Can anyone please clarify this situation - preferably formally and explicitly?

    Peter
     

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    Last edited: Jul 7, 2012
  2. jcsd
  3. Jul 7, 2012 #2

    lavinia

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    A loop is a continuous map from a circle into a space. If f is a loop in X and if h is continuous then hf is a continuous map from the circle into Y and is therefore a loop.

    The fundamental group is constructed from loops that begin and end at a fixed given point, the so called base point. It h preserves base points then a loop at the base point in X will be mapped to a loop at the base point in Y.

    One can consistently define where a loop,f, begins and ends by thinking of it as a map of the unit interval into a space whose value is the same at 0 and 1. Then by definition the loop begins and ends at f(0),
     
  4. Jul 7, 2012 #3
    Thanks Lavinia

    OK so after we are given that base point is preserved, it is continuity that ensures we have a loop in Y

    Thanks again

    Peter
     
  5. Jul 7, 2012 #4

    lavinia

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    right
     
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