TITLE: "Why the heck are the homotopy groups of the sphere so damn complicated?" 
QUESTION [50 upvotes]: This is a quote from a dear friend asking the rest of us on Facebook. I gave him some half-baked response, but the truth is I don't really know enough about this to give him a good response. 
So why ARE they so complicated? The topologists here want to give a few responses so I can give him some feedback to his desperate query?

REPLY [11 votes]: I might be misinterpreting your friend's question, but if your friend knows very little about the subject and is expressing surprise that something so seemingly simple turns out to be so complicated, then maybe you should concentrate on explaining the Hopf fibration to him or her.  There is an interesting visualization in Thurston and Levy's book Three-Dimensional Geometry and Topology.  I think that anyone who successfully goes through the exercise of visualizing the Hopf fibration will be disabused of any preconception that the homotopy groups "should" be simple.