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- W2050878853 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." @default.
- W2050878853 created "2016-06-24" @default.
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- W2050878853 date "1987-02-01" @default.
- W2050878853 modified "2023-10-02" @default.
- W2050878853 title "About the extension problem for proper maps" @default.
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- W2050878853 doi "https://doi.org/10.1016/0166-8641(87)90074-5" @default.
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