Geodesic
Obstructions to the extension of partial maps
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 803-812.

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One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map $Z\hookleftarrow A\overset{f}{\to} X$, i.e., of a subset $F\subset Z\setminus A$ such that $f$ can be globally extended to its complement. It is shown that if $Z$ is a fixed metric space with $\dim Z\le n$ and $p,q\ge-1$ are fixed numbers, then obstructions to extending all partial maps $Z\hookleftarrow A\overset{f}{\to} X\in\operatorname{LC}^p\cap \operatorname{C}^q$ can be concentrated in preselected fairly thin subsets of $Z$. 
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S. M. Ageev; S. A. Bogatyi. Obstructions to the extension of partial maps. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 803-812. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a0/

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