Atlas: Resolutions for Metrizable Compacta in Extension Theory by Leonard R. Rubin

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Geometric Topology II
September 29 - October 5, 2002
Inter-University Center, Dubrovnik; Department of Mathematics, University of Zagreb
Dubrovnik, Croatia

Organizers
Ivan Ivansic, University of Zagreb;, James E. Keesling, University of Florida;, Alexander N. Dranishnikov, University of Florida;, Sime Ungar, University of Zagreb

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Resolutions for Metrizable Compacta in Extension Theory
by
Leonard R. Rubin
University of Oklahoma
Coauthors: Philip J. Schapiro

We shall speak about a K-resolution theorem for simply connected CW-complexes K in extension theory in the class of metrizable compacta X. This means that if dimX <= K (in the sense of extension theory), n is the first element of \N such that G=\pi_n(K) =/= 0, and it is also true that \pi_n+1(K)=0, then there exists a metrizable compactum Z and a surjective map \pi:Z\ra X such that:

": (a)" \pi is G-acyclic,
": (b)" dimZ <= n+1, and
": (c)" dimZ <= K. If additionally, \pi_n+2(K)=0, then we may improve (a) to the statement,
": (aa)" \pi is K-acyclic. To say that a map \pi is K-acyclic means that each map of each fiber \pi^-1(x) to K is nullhomotopic.

In case \pi_n+1(K) =/= 0, we obtain a resolution theorem with a weaker outcome. Nevertheless, it implies the G-resolution theorem for arbitrary abelian groups G in cohomological dimension dim_G X <= n when n >= 2.

Date received: July 23, 2002