Collapsing successors of singular cardinals using a theorem of Magidor

In their paper [1] Adolf, Apter and Koepke showed that by assuming enough super-compactness for kappa one can prove that Aleph_{w1 + 1} can be collapsed to have cofinality w1 while preserving Aleph_{w1} and all cardinals above Aleph_{w1 + 1}. For this, they use Magidor's work [2] to get the required two-stage forcing.

The main goal of the lecture is to address the following natural question: When does this forcing preserve cardinals below kappa? We shall prove that no new bounded subsets of kappa are introduced by the forcing if and only if the cofinality of kappa is countable.

References
[1] Adolf, Apter, and Koepke (2017). Singularizing successor cardinals by forcing. Proc. Amer. Math. Soc. 146 (2018), 773-783.
[2] Magidor, M. (1977). On the singular cardinals problem i. Israel Journal of Mathematics.