Highness, locally noncappability and nonboundings

In this paper, we improve a result of Seetapun and prove that above any nonzero, incomplete recursively enumerable (r.e.) degree a, there is a high2 r.e. degree c>a witnessing that a is locally noncappable (Theorem 1.1). Theorem 1.1 provides a scheme of obtaining high2 nonboundings (Theorem 1.6), as all known high2 nonboundings, such as high2 degrees bounding no minimal pairs, high2 plus-cuppings, etc.