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 $q$

Asymptotic topology

We establish some basic theorems in dimension theory and absolute extensor theory in the coarse category of metric spaces. Some of the statements in this category can be translated in general topology language by applying the Higson corona functor. The relation of problems and results of this `Asymptotic Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe

On Bestvina-Mess Formula

Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$\dim_L\partial\Gamma=cd_L\Gamma-1$$ connecting the cohomological dimension of a group $\Gamma$ with the cohomological dimension of its boundary $\partial\Gamma$. In [Be] Bestvina introduced a notion of \sZ-structure on a discrete group and noticed that his formula holds true for all torsion free groups with \sZ-structure. Bestvina's notion of \sZ-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivially is not valid for groups with torsion, we show that it still holds in the following modified form: {\it The cohomological dimension of a \sZ-boundary of a group $\Gamma$ equals its global cohomological dimension for every PID $L$ as the coefficient group} $$\dim_L\partial\Gamma=gcd_L(\partial\Gamma).$$ Using this formula we show that the cohomological dimension of the boundary $\dim_{L}\partial\Gamma$ is a quasi-isometry invariant of a group.Comment: 10 page