Let $X$ be a finite simplicial complex such that its geometric realisation $|X|$ is homeomorphic to a closed, connected, orientable manifold $M^n$. Suppose we are given an element $\gamma \in \pi_1(|X|)$ (where we know the fundamental group from abstract, non-implemented reasons).
My Question: Is there a sort of algorithm that allows us to construct a representative of this homotopy class (maybe even as a simplicial embedding)?
I do not know how to think about this problem. I think that an analogous statement for (simplicial) homology works if one supposes that the Hurewicz homomorphism is surjective and the dimension is high enough (i.e., we want to find an embedding $\partial \Delta^2 \hookrightarrow |X|$ that represents a given homology class, where we know that a suitable embedding exists in $M$ via abstract reasons): Then one can just write down every simplicial embedding of $\partial \Delta^2$ into the complex (maybe iteratively subdivide), which is possible since all complexes involved are finite. Since (simplicial) homology can be computed algorithmically, one can check if the result is the desired class and by the abstract existence of a suitable embedding, this algorithm terminates.
However, these techniques do not seem to work for homotopy groups since they can be not computed algorithmically (or at least I do not know how it could be done). Any help, literature or helpful thoughts are greatly appreciated!