  The aim of this paper is to show the importance of the Steenrod construction
of homology theories for the disassembly process in surgery on a generalized
$n$-manifold $X^n$, in order to produce an element of generalized homology
theory, which is basic for calculations. In particular, we show how to
construct an element of the $n$-th Steenrod homology group $H^{st}_{n} (X^{n},
\mathbb{L}^+),$ where $\mathbb{L}^+$ is the connected covering spectrum of the
periodic surgery spectrum $\mathbb{L}$, avoiding the use of the geometric
splitting procedure, which is standardly used in surgery on topological
manifolds.
