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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.