Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fundamental group question

  1. Mar 7, 2007 #1


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Simple question: is the fundamental group of a pointed space independant of the base point?
  2. jcsd
  3. Mar 7, 2007 #2


    User Avatar
    Homework Helper

    If the space is path connected. Then if [itex]\gamma[/itex] is a path from x1 to x0, the map sending the homotopy class of a loop [itex]\alpha[/itex] in [itex]\pi_1(X,x_0)[/itex] to the homotopy class of the loop [itex]\gamma \alpha \gamma^{-1} [/itex] in [itex]\pi_1(X,x_1)[/itex]is easily shown to be an isomorphism.
  4. Mar 7, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    but a different path can yield a different isomorphism. hence the two groups are isomorphic but there is no distionguished isomorphism. so the two groups cannot be identified, so the answer is no, the group itself depends on the point, but the isomorphism class does not.

    we often think iof isomorphic groups as "the same" but of course they are not, else there would be no galois theory.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Fundamental group question
  1. Group question (Replies: 19)

  2. Cyclic group question (Replies: 2)