In a recent paper I wrote with John Oprea (Oprea, John; Strom, Jeff Lusternik-Schnirelmann category, complements of skeleta and a theorem of Dranishnikov. Algebr. Geom. Topol. 10 (2010), no. 2, 1165–1186) we proved that if $X$ is an $n$-dimensional simplicial complex, then the complement of the $r$-skeleton $X_r$ is a complex $Y_{n-r-1}$ of dimension at most $n-r-1$ and there is another subcomplex $L$ of $X$ such that 
$$
\begin{array}{ccc}
L&\rightarrow &X_{r}\\
\downarrow&&\downarrow\\
Y_{n-r-1}&\rightarrow &X_n
\end{array}
$$
is a homotopy pushout square.