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Leonard R. Rubin Philip J. Schapiro 《Transactions of the American Mathematical Society》2006,358(6):2507-2536
We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:
(a) is -acyclic,
(b) , and
(c) .
This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .
If in addition , then (a) can be replaced by the stronger statement,
(aa) is -acyclic.
To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.