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.
Homology groups can be computed with Smith normal form (see this survey). 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). The same paper shows that it is NP-hard to decide if a simplicial complex is collapsible (a condition which implies contractible).