The polynomial time hierarchy collapses if the Boolean hierarchy collapses

It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that co-NP⊆NP^S, and therefore the polynomial-time hierarchy (PH) collapses to a subclass of ΔP/3. Since the BH is contained in P^NP, these results relate the internal structure of P^NP to the structure of the PH as a whole. Other conditions that imply the collapse of the BH (and the collapse of the PH in turn) are examined