PATCHING and the COMPLETED HOMOLOGY of LOCALLY SYMMETRIC SPACES

Under an assumption on the existence of -adic Galois representations, we carry out Taylor-Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply 'big ' theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where and splits completely in the number field, we relate our construction to the -adic local Langlands correspondence for.