Cell-like resolutions preserving cohomological dimensions

We prove that for every compactum X with dim_Z X = 2 there is a cell-like resolution r: Z --> X from a compactum Z onto X such that dim Z <= n and for every integer k and every abelian group G such that dim_G X = 2 we have dim_G Z <=k. The latter property implies that for every simply connected CW-complex K such that e-dim X <= K we also have e-dim Z <= K.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-46.abs.htm

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

The LS category of the product of lens spaces

We reduced Rudyak's conjecture that a degree one map between closed manifolds cannot raise the Lusternik-Schnirelmann category to the computation of the category of the product of two lens spaces $L^n_p\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat(L^n_p\times L^n_q)$ for values of $p,q>n/2$. It turns out that our computation supports the conjecture. For spin manifolds $M$ we establish a criterion for the equality $cat M=dim M-1$ which is a K-theoretic refinement of the Katz-Rudyak criterion for $cat M=dim M$. We apply it to obtain the inequality $cat(L^n_p\times L^n_q)\le 2n-2$ for all $n$ and odd relatively prime $p$ and $q$