Talk:Topology/The fundamental group

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I do not understand the sentence "This section is dedicated to the calculation of the fundamental group of S1 that we can consider contained in the complex topological space". Is "complex topological space".the space of complex numbers? Why not use RxR? Would the topology be any different?

Is it true that the specialty of S1 (compared with S2 or higher dimensions) is that there is not enogh "room" for a "film" between loops with different numbers of turns?

Artie01: Hi! I'm not the author of the above sentence, but I think I can explain it a bit. By the "complex topological space" I presume he/she means the complex numbers C, equipped with its standard topology. (Remember that stricly speaking, you can't just specify the underlying set --- you have to specify the topology on that set too.) As you suggest, there's no practical difference between C and RxR as topological spaces (when both have their usual topologies). I'd guess that the author discusses things in terms of C since the proof of the main theorem --- \pi_1 S^1 is isomorphic to Z --- views S^1 as the unit circle in C.

As for your second question: there are lots of intuitive ways to view the difference between the fundamental group of S1 and those of all other spheres. I like to think about a loop of stretchy string that's constrained to lie on the surface of S^n. For n > 1, it's clear that, no matter how crazy the loop is, we can always contract it down to a single point. But on S^1, no matter how we twist or stretch the string we can never unwind it.

Hope this helps...