A relationship between difference hierarchies and relativized polynomial hierarchies

Chang and Kadin have shown that if the difference hierarchy over NP collapses to level $k$, then the polynomial hierarchy (PH) is equal to the $k$th level of the difference hierarchy over $\Sigma_{2}^{p}$. We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level $k$, then PH = $\left(P_{(k-1)-tt}^{NP}\right)^{NP}$. We also extend the result to classes other than NP: For any class $C$ that has $\leq_{m}^{p}$-complete sets and is closed under $\leq_{conj}^{p}$and $\leq_{m}^{NP}$-reductions, if the difference hierarchy over $C$ collapses to level $k$, then $PH^{C} = $\left(P_{(k-1)-tt}^{NP}\right)^{C}$. Then we show that the exact counting class $C_{=}P$ is closed under $\leq_{disj}^{p}$and $\leq_{m}^{co-NP}$-reductions. Consequently, if the difference hierarchy over $C_{=}P$ collapses to level $k$ then $PH^{PP}$ is equal to $\left(P_{(k-1)-tt}^{NP}\right)^{PP}$. In contrast, the difference hierarchy over the closely related class PP is known to collapse. Finally, we consider two ways of relativizing the bounded query class $P_{k-tt}^{NP}$: the restricted relativization $P_{k-tt}^{NP^{C}}$, and the full relativization $\left(P_{k-tt}^{NP}\right)^{C}$. If $C$ is NP-hard, then we show that the two relativizations are different unless $PH^{C}$ collapses.