Positive scalar curvature and strongly inessential manifolds

We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M$ which is strongly inessential. The latter means that a classifying map $u:Mto K(pi_1(M),1)$ can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds with the exception of 4-manifolds with spin universal coverings.

On Iwases manifolds

In ~cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and $M_3$ with LS-category 3 which are counter-examples to Ganeas conjecture: ${rm cat_{LS}} (Mtimes S^n)={rm cat_{LS}} M+1$. We show that the manifold $M_3$ is a counter-example to the logarithmic law for the LS-category of the square of a manifold: ${rm cat_{LS}}(Mtimes M)=2{rm cat_{LS}} M$. Also, we construct a map of degree one $$f:Nto M_2times M_3$$ which reduces Rudyaks conjecture to the question whether ${rm cat_{LS}}(M_2times M_3)ge 5$. We show that ${rm cat_{LS}}(M_2times M_3)ge 4$.

Surgery Approach to Rudyaks Conjecture

Using the surgery we prove the following: THEOREM. Let $f:M to N$ be a normal map of degree one between closed manifolds with $N$ being $(r-1)$-connected, $rge 1$. If $N$ satisfies the inequality $dim N leq 2r cat N - 3$, then for the Lusternik-Schnirelmann category $cat M geq cat N$ .

On Macroscopic dimension of non-spin 4-manifolds

We prove that for 4-manifolds $M$ with residually finite fundamental group and non-spin universal covering $Wi M$, the inequality $dim_{mc}Wi Mle 3$ implies the inequality $dim_{mc}Wi Mle 2$.

On dimension of product of groups

We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.

The topological complexity of the free product

We prove the formula $TC(Gast H)=max{TC(G), TC(H), cd(Gtimes H)}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.

On LS-category and topological complexity of connected sum

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $cat(M# N)=max{cat M,cat N}$. For topological complexity we prove the inequality $TC (M# N)gemax{TC M,TC N}$ for simply connected manifolds.

Embedding of Coxeter groups in a product of trees

We prove that a right angled Coxeter group with chromatic number n can be embedded in a bilipschitz way into the product of n locally finite trees. We give applications of this result to various embedding problems and determine the hyperbolic rank of products of exponentially branching trees.