About the extension problem for proper maps

Abstract

In this paper, we study the extension problem in the category of topological spaces and proper maps. To attack this problem a new proper cohomology theory and a new obstruction cocycle are defined. This cohomology theory has coefficients in a morphism π′ → π where π′ is a pro-abelian group and π is an abelian group. Let K n be the n-skeleton of a second countable, locally compact cell complex K, and let Y be a topological space with a cofinal sequence of compact subsets Ø = M 0 ⊂ M 1 ⊂ M 2 ⊂⋯⊂ Y such that Y − M i is a path-connected n-simple space. In this case, the sequence ⋯→ π n(Y − M 2) → π n(Y − M 1) → π n(Y) can be seen as a morphism of the pro-abelian group π′ = { π n ( Y − M i )| i⩾1} to the abelian group π = π n ( Y). Then we define an obstruction cocycle c n+1 ( g) with coefficient in π′ → π and prove the following results. (Proposition) A proper map g: K n → Y has a proper extension over K n+1 if and only if c n+1 ( g) = 0. (Theorem) Let g: K n → Y be a proper map. Then g/ K n–1 can be a properly extended over K n+1 if and only if c n+1 ( g) is cohomologous to zero.