To compute the simplicial homotopy group of a space $X$, we find a Kan fibrant replacement $X \to Y$ and calculate for that for $Y$, which can be implemented in a computer program.


Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.


Could you please show why it's hard, for the easiest possible nontrivial case?



Kan fibrant replacement for a sphere

> 
> Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.
> 
> 
> 


Computing the fibrant replacement for simplicial sets is quite
easy: it is given by the [Kan fibrant replacement functor](https://ncatlab.org/nlab/show/Kan+fibrant+replacement) Ex^∞. Explicitly, n-simplices in the fibrant replacement of a simplicial set X are maps Sd^k Δ^n → X, for some k≥0.
Here Sd^k denotes the k-fold barycentric subdivision of a simplicial set.
We allow to increase k by further subdividing, this does not change the simplex.


This description is very simple and can be easily programmed into a computer.


The problem is, however, is that the number of simplices grows exponentially
with k, and we also do not have an efficient way to get an a priori upper
bound for k. So some problems are bound to be computationally undecidable,
such as the problem of computing whether π\_1 of a simplicial set is trivial or not.


