This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called.

Take an $n$ dimensional simplicial complex $X$ and define a new simplicial complex $X_{k}$ in the following way – the vertices of $X_{k}$ will be $k$ dimensional simplices of $X$ and two such vertices will be connected by an edge if the two corresponding $k$ simplices in $X$ are contained in a $k+1$ simplex in $X$. This is the skeleton of $X_{k}$ and complete it by gluing higher dimensional simplices whenever they are formed (so you get an $n-k$ simplicial complex).

My questions – what is the standard name for this and where can I find a reference on? Moreover, is it true that whenever $X$ is contractible then $X_k$ is simply connected (and even that the homologies of order < $n-k$ vanish)?