Universal acyclic resolutions for arbitrary coefficient groups

We prove that for every compactum $X$ and every integer $n \geq 2$ there are a compactum $Z$ of $\dim \leq n+1$ and a surjective $UV^{n-1}$-map r: Z \lo X such that for every abelian group $G$ and every integer $k \geq 2$ such that $\dim_G X \leq k \leq n$ we have $\dim_G Z \leq k$ and $r$ is $G$-acyclic

Rational acyclic resolutions

Let X be a compactum such that dim_Q X 1. We prove that there is a Q-acyclic resolution r: Z-->X from a compactum Z of dim < n+1. This allows us to give a complete description of all the cases when for a compactum X and an abelian group G such that dim_G X 1 there is a G-acyclic resolution r: Z-->X from a compactum Z of dim < n+1.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-12.abs.htm

Maps to the projective plane

We prove the projective plane \rp^2 is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space \Map(\rp^n,\rp^{n+1}) (based at inclusion) as being isomorphic to either $\Z_4$ or $\Z_2\oplus\Z_2$ for $n\ge 1$. Double surgery and the above fact yield the proof.Comment: 17 page