My professor told me that computing higher homotopy groups $\pi_n(X)$ for $n\geq 2$ of a space $X$ can be rather complicated, and one wants to use homology instead.
I find in this link that the higher homotopy groups of the spheres have been computed.
Which techniques were used, or are they are computed in a brute force manner?
Best Answer
The higher homotopy groups of spheres are far from known. In the unstable range not much is computed above the $n+20$ to $n+30$ stem. There are techniques to compute these things but mostly these need to be applied on a case by case basis. Even then you still run into group extension problems that may or may not realistically be solvable. You can read a little bit here https://mathoverflow.net/questions/190965/unstable-homotopy-groups-of-spheres-beyond-todas-range on what is still more or less state of the art.
The Japanese mathematician H. Toda is responsible for much of what is known, and his beautiful, but perhaps difficult, memoir Composition Methods in Homotopy Groups of Spheres is the go to place to start learning about how exactly these things can be computed.
Stably the story is different and a lot more is known. We are still operating on a case by case basis, but here the scope of each case is a lot wider. What we are doing here is computing differentials in the Adams spectral sequence. Once you compute a single differential you get a whole lot more information rather than just a single group. However you soon run into the problem of computing the next differential, and you end up back where you started. I have no idea of exactly how far the computations of the stable groups run, so if anyone does know of the state of the art do let us know.
A good reference for the Adams-Novikov spectral sequence comes in the form of D. Ravenel's book Complex Cobordism and Stable Homotopy Groups of Spheres. He's also written another book on the subject, Nilpotence and Periodicity in Stable Homotopy Theory, which is a good reference for related material. These two books are known as the green and orange books, respectively, and are available on his website.