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)? 



The "grassmannian" of a simplicial complex
The claims about simple connectivity and homology vanishing in the last paragraph is false. 


For a failure of simple connectivity, let $X$ be the two dimensional simplicial complex with vertex set $\{ a,b,c,d,e \}$ and maximal faces $abe$, $bce$, $cde$, $ade$. This is contractible: It is a solid square subdivided into four triangles.


Then $X\_1$ retracts onto the circle whose vertices are $ae$, $be$, $ce$ and $de$. 


For a failure of homology vanshing, let $X$ be the three dimensional simplicial complex with vertex set $\{ a,b,c,d, p,q,r,s,x \}$ and maximal faces $abpx$, $bpqx$, $bcqx$, $cqrx$, $cdrx$, $drsx$, $adsx$, $apsx$. So we have triangulated a cylinder and coned it from the point $x$. Since there is a cone point, the complex is contractible. 


Each tetrahedron of $X$ contributes a hollow octahedron to $X\_1$, and they are glued together into a ring. So $X\_1$ is homotopic to a wedge of $S^1$ and eight $S^2$'s, and in particular $H^1(X\_1) \neq 0$. 


