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# Extension of Maps to Nilpotent Spaces

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Published:2001-09-01
Printed: Sep 2001
• M. Cencelj
• A. N. Dranishnikov
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## Abstract

We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.
 Keywords: cohomological dimension, extension of maps, nilpotent group, nilpotent space
 MSC Classifications: 55M10 - Dimension theory [See also 54F45] 55S36 - Extension and compression of mappings 54C20 - Extension of maps 54F45 - Dimension theory [See also 55M10]

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