# Question about collapsing cardinals

Suppose, in $M$, $\kappa$ regular, $\lambda>\kappa$ regular. Is there a generic extension of $M$ in which $\kappa^+ = \lambda$ and in which cardinals $\leq \kappa$ and $\geq \lambda$ are preserved?

I worked out that, assuming GCH, the answer is yes if $\lambda$ is a limit or is the successor of a cardinal of cofinality $\geq\kappa$. The only remaining case is $\lambda = \delta^+$ for some $\delta$ of cofinality $<\kappa$, e.g. $\kappa = \omega_1$ and $\lambda = \omega_\omega^+$.

I realize that in this remaining case, in $M[G]$ $\kappa^{<\kappa} \geq \lambda$, so the forcing notion cannot be $<\kappa$-distributive. Thus the Levy collapse cannot suffice.

Many questions of this sort are open. For example, consider the situation where $\kappa = \aleph_n$ and $\lambda = \aleph_{n+2}$. So you want to know if you can collapse $\aleph_{n+1}$ to $\aleph_n$ while preserving all other cardinals. Now if $n=0$ this is possible thanks to the Levy collapse. For $n=1$, this is again possible, which is due independently to Abraham and Todorcevic. For $n=2$ this was recently answered by Aspero (you will also find references to the other articles there). For $n \geq 3$ this is still open, and I believe it is also open, for example, whether you can collapse $\aleph_{\omega+1}$ to $\aleph_1$ with a stationary set preserving partial order of size $\aleph_{\omega+1}$ (which would hence not collapse any larger cardinals and also preserve $\aleph_1$).