Simply Connected and Fundamental Group

  • Thread starter waht
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Main Question or Discussion Point

I have a little hard time understanding the definition of a simply connected space in terms of a fundamental group. A space is simply connected if its fundamental group is trivial, has only one element?

It's been some time since I played around with homotopy. My understanding is that a set of path homotopy classes of loops satisfies the axioms of a group.

Is one element of that group just one loop? If so, how does that tell you there is no holes the loop is encompassing. This where I get confused. Thanks.
 

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  • #2
AKG
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The fundamental group consists of homotopy classes of loops, so there is only one class of loops, not only one loop. The fact that there is only one class of loops means that every loop in your space is in that single class. This means every loops is homotopic to every other loop. There are always loops encompassing no holes (e.g. single-point loops). If there were also a loop encompassing some hole, then this hole-encompassing loop would be homotopic to a non-hole-encompassing loop, which can't be. So there are no hole-encompassing loops.
 
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Thanks that cleared thing up.
 

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