Resolution and Product Theorems for Cohomological Dimension
by
Philip Schapiro
Langston University
Coauthors: Leonard R. Rubin

Limit Theorem | -Inv -sh -Fd

5.5.5

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esolution and Product Theorems for Cohomological Dimension

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eonard R. Rubin and Philip J. Schapiro

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epartment of Mathematics, The University of Oklahoma, 601 Elm Avenue, Room 423, Norman, OK 73019 U.S.A. lrubin.edu

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epartment of Mathematics, Langston University, Langston, OK 73050 U.S.A. schapiro1.lunet.edu Let X be a metrizable space and n >= 0. We show that there exist completely metrizable spaces [X\tilde], [Z\tilde] with X densely embedded in [X\tilde], dim[Z\tilde] <= n, and a proper surjective \operatornameUV^n-1-map [(\pi)\tilde]:[Z\tilde] --> [X\tilde]. In case dim_Z X <= n, we construct this resolution [(\pi)\tilde] so that it is a cell-like map. On the other hand, if G is an abelian group which is the direct sum of groups of the form Z/p^k where p is a prime and k in N, and dim_G X <= n, then we obtain the resolution [(\pi)\tilde] so that in addition to being \operatornameUV^n-1, it is Z/p^k-acyclic for each such p^k. All the spaces X, [X\tilde], [Z\tilde] and Z=[(\pi)\tilde]^-1(X) are of the same weight. There is a product theorem:

dimG X×Y <= dimG X+dimG Y

if G is a direct sum of groups of the type Z or Z/p^k. We, moreover, have a completion theorem. Namely, if G is of the preceding type, then there exists a completely metrizable space [X\tilde] containing X with dim_G[X\tilde]=dim_GX. In this restricted class of groups G, each metrizable space has a Bockstein type basis which is uncovered via a theory of extension types.

Date received: January 8, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-05.