Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell whether a given simplical complex is contractible by showing that the fundamental group and all reduced homology groups vanish, but if there is a faster way to compute whether a simplicial complex is contractible then that would be helpful as well. 



Computational complexity of computing simplicial homology
Homology groups can be computed with Smith normal form (see [this survey](http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/Publications/DHSW.pdf)). As for deciding if a simplicial complex is contractible, that is difficult. It is undecidable to tell if a simplicial complex is contractible (see appendix A of [this paper](http://arxiv.org/abs/1211.6254v1)). The same paper shows that it is NP-hard to decide if a simplicial complex is [collapsible](https://en.wikipedia.org/wiki/Collapse_(topology)) (a condition which implies contractible).


