I am a new learner in topology and manifold and get some problems.
The lemma that higher homotopy group of a pathwise connected topological space are commutative can be proved by using Eckmann-Hilton argument.
However, I didn't get a intuitive picture about how it's commutative. What's the essentially difference between higher homopy group and fundamental group that cause commutation or not.
Could anyone help me with that?
Answering to question asking for intutions is always hard, especially because different people can have differnt intuitions for the same thing, and different intutions may not work for someone.
With that said here my 2 cents.
The trick of Eckmann-Hilton argument is the fact that we have two composition laws on higher paths, that commute with each other and that have the same identities.
If you look at higher-paths as maps from $n$-cubes into the space the compositions laws correspond to gluing the cubes along two different directions.
Using the identities allows you to deform a composition of two $n$-cubes in a composition of four n-cubes (the two cubes and two copies of the identity cube).
Then the commutativity of the two operations allows to rotate these $n$-cubes inside the larger composite cube while not altering the result of the composition.
Unfortunately I cannot attach the drawing that would explain better what I have said, but I suggest you to look at Hatcher's Algebraic Topology where you can find a proof with drawings that explain what I just said.
I hope this helps.