Conservation theorems for the Cohesiveness Principle

We prove that the Cohesiveness Principle (COH) is $\Pi^1_1$ conservative over $RCA_0 + I\Sigma^0_n$ and over $RCA_0 + B\Sigma^0_n$ for all $n \geq 2$ by recursion-theoretic means. We first characterize COH over $RCA_0 + B\Sigma^0_2$ as a `jumped' version of Weak K\"{o}nig's Lemma (WKL) and develop suitable machinery including a version of the Friedberg jump-inversion theorem. The main theorem is obtained when we combine these with known results about WKL. In an appendix we give a proof of the $\Pi^1_1$ conservativity of WKL over $RCA_0$ by way of the Superlow Basis Theorem and a new proof of a recent jump-inversion theorem of Towsner.