I've seen some works on the representation of fundamental groups, which are (at least for me) quite important topic in mathematics. For example, Riemann-Hilbert correspondence relates representation of a fundamental group of complex algebraic variety with differential equations on it. (There's also a result in positive characteristic by Bhatt-Lurie). Also, etale fundamental group and its representation is also important in number theory because it is closely related to Galois groups. (I don't know much about this though). However, I can't find any works that studies about representation of higher homotopy groups $\pi_{n}$ (or etale homotopy groups, whatever it is). Although $\pi_n$ is abelian for $n\geq 2$, there might exist some nontrivial stuff that $\pi_n$ can act naturally so that we can study about its representations (which would be 1-dimensional if irreducible, though). Could you give any examples if there's any interesting works about such things?

  • $\begingroup$ There is a natural action of fundamental group on higher homotopy groups. I don't know how a higher homotopy group could naturally act on another higher homotopy group. $\endgroup$
    – Z. M
    Jun 6, 2021 at 11:31
  • 6
    $\begingroup$ I think that a "more correct" thing would be to consider the whole homotopy type (it incorporates pi_1 but also pi_2 and so on). So you want a "homotopy group" to act on something. The clearest thing conceptually would be to think of local systems. Local system of what? Depending on the answer to that question, you suddenly might start to see the need for pi_2 etc. $\endgroup$
    – Sasha
    Jun 6, 2021 at 13:16
  • 2
    $\begingroup$ @Z.M The usual action of a higher homotopy group of a space on the higher homotopy groups of that same space is by the Whitehead product: en.wikipedia.org/wiki/Whitehead_product $\endgroup$
    – user164898
    Jun 6, 2021 at 17:50

1 Answer 1


There are many results that generalize the Riemann–Hilbert correspondence from the fundamental groupoid to the fundamental ∞-groupoid, for example:

For the higher Galois theory, see

  • $\begingroup$ Thank you for your reply! I'll try to read. $\endgroup$
    – Seewoo Lee
    Jun 9, 2021 at 2:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.