Simply Connected and Fundamental Group

In summary, a space is simply connected if its fundamental group has only one homotopy class of loops, meaning that every loop in the space is homotopic to every other loop. This means there are no loops encompassing holes in the space.
  • #1
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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
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.
 
  • #3
Thanks that cleared thing up.
 

FAQ: Simply Connected and Fundamental Group

1. What is the definition of a simply connected space?

A simply connected space is a type of topological space where any loop can be continuously shrunk to a point without leaving the space. In other words, there are no holes or gaps in the space that cannot be "filled in" by a continuous path.

2. How is a simply connected space different from a path-connected space?

A path-connected space is one where there exists a continuous path between any two points in the space. Simply connected spaces take this a step further by requiring any loop to also be continuously shrinkable to a point. Therefore, all simply connected spaces are also path-connected, but not all path-connected spaces are simply connected.

3. What is the fundamental group of a simply connected space?

The fundamental group of a simply connected space is the trivial group, which contains only one element (the identity element). This means that all loops in the space are homotopic to a point, and therefore have a trivial group action.

4. Can a space be simply connected but not contractible?

Yes, a space can be simply connected but not contractible. A contractible space is one that can be continuously deformed to a point, which is a stronger condition than simply connectedness. An example of a simply connected but not contractible space is the unit circle in the plane.

5. How can the fundamental group be used to classify topological spaces?

The fundamental group is a topological invariant, meaning that it does not change under homeomorphisms (continuous, one-to-one maps with a continuous inverse) of a space. This allows for different spaces to be classified based on the isomorphism classes of their fundamental groups. For example, any two spaces with a fundamental group isomorphic to the integers are considered topologically equivalent.

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