Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Z \leftarrow A\xrightarrow{f}X$$ \end{document}, i.e., of a subsetF ⊂ Z/A such thatf can be globally extended to its complement. It is shown that ifZ is a fixed metric space with dimZ ≤ n andp, q ≥−1 are fixed numbers, then obstructions to extending all partial maps\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Z \leftarrow A\xrightarrow{f}X \in LC^p \cap C^4 $$ \end{document} can be concentrated in preselected fairly thin subsets ofZ.