Wikipedia Trails From Fundamental Group to N-Sphere

Fundamental Group

Something I have been interested in ever since I started learning about topology was the fundamental group. So, I decided to start my Wikipedia trail here. According to Wikipedia, the fundamental group of X at the point x_o is, "...the set of all homotopy classes of loops with base point x_o..." This likely doesn't make sense to most others, but it does to me... except for one small definition.

Homotopy

My next Wikipedia article was about homotopies because I didn't know what they were. They make sense to me now, fortunately. Essentially, you take two paths in space between a point A and B, and a homotopy between the two paths is a way of deforming one path into the other. Here's a beautiful gif, courtesy of Wikipedia:

Caption: "The two dashed paths shown above are homotopic relative to their endpoints." Source: Wikimedia

Homotopy Group

The article on homotopy led me to the homotopy group, which I thought was a natural next step. However, little did I know what I was getting into. I believe the homotopy group to just be a set of homotopy classes of maps, but from a particular n-sphere to a space X. It was kind of confusing, I will have to study this a little more.

N-Sphere

I decided to end my Wikipedia venture on the n-sphere. It's a rather simple object, but seems to have great importance in algebraic topology, surprisingly. The n-sphere is defined to be the set of all points whose distance from a particular point is constant. Interestingly, the n-sphere has one dimension less than the ambient space it is in. For instance, the 1-sphere is a circle in the two-dimensional plane, while the 2-sphere is a literal sphere in three-dimensional space.