# Computational complexity of computing simplicial homology

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.

• I added the [reference-request] tag. – David Roberts Jun 30 '16 at 4:37
• Note that your context includes the undecidable problem of determining in general whether a group is trivial given a (finite) presentation of it. (This is a particular case of the group isomorphism problem that is also known to be undecidable.) – Eric Towers Jun 30 '16 at 5:56