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Consider some space $X$. Does the fundamental group tell us information about the equivalence between two paths $f,g: I \to X$? So there exists a homotopy $h: I \times I \to X$ such that $h(s,0) = f(s)$, $h(s,1) = g(s)$, $h(0,t) = x$ and $h(1,t) = y$?

In other words, starting with two paths, how/why do we introduce the fundamental group?

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    Have you tried reading a textbook on the subject? There are great expositions! The end result will be much, much better than even the best answer you might get here.2011-06-17
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    I'm not sure I understand your question. First of all, by equivalent do you mean homotopic, as your second question seems to imply?2011-06-17
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    @Becca: Yes. (more characters)2011-06-17
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    Oh, maybe a better question for me to ask: do you mean homotopic relative to the boundary of the paths where the boundary is the endpoints x and y?2011-06-17

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