The Eilenberg–Borsuk theorem for mappings into an arbitrary complex

The classical Eilenberg–Borsuk theorem on extension of partial mappings into a sphere is generalized to the case of an arbitrary complex $K$. It is formulated in terms of extraordinary dimension theory, which is developed in the present paper. When $K = K(G,\, k)$ is an Eilenberg–MacLane complex, the result can be expressed in terms of cohomological dimension theory. For partial mappings $\varphi\colon A\to K(G,\, k)$ of an $n$-manifold $M$, the following is obtained:
Theorem. If $k<n-2$, then there exists a compactum $X\subset M$ of dimension $n-k-1$, such that the mapping $\varphi$ extends to $M-X$ and for every abelian group $\pi$ with $\pi\otimes G=0$ the cohomological dimension of $X$ with coefficients in $\pi$ does not exceed $n-k-2$.
Thus, in comparison with the classical Eilenberg–Borsuk theorem, there is obtained an additional condition as to the cohomological dimension of $X$.