On topological complexity of hyperbolic groups
We show that the topological complexity of a finitely generated torsion free hyperbolic group $π$ with $\cdπ=n$ equals $2n$.
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arXiv subjects
Alexander Dranishnikov
Publications and source records attributed to Alexander Dranishnikov.
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 $\dim_{mc}\Wi M\le 3$ implies the inequality $\dim_{mc}\Wi M\le 2$.
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An upper bound on the LS-category in presence of the fundamental group
We prove that $$ \cat X\le \frac{\cd(π_1(X))+\dim X}{2}$$ for every CW complex $X$ where $\cd(π_1(X))$ denotes the cohomological dimension of the fundamental group of $X$. We obtain this as a corollary of the inequality $$ \cat X\le\frac{\cat (u_X)+\dim X}{2} $$ where $u_X:X\to Bπ_1(X)$ is a classifying map for the universal covering of $X$.
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The topological complexity of the free product
We prove the formula $TC(G\ast H)=\max\{TC(G), TC(H), cd(G\times H)\}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.
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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 $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.
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Asymptotic dimension, decomposition complexity, and Haver's property C
The notion of the decomposition complexity was introduced in \cite{GTY} using a game theoretical approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C. Then we define a game theoretical analog of Haver's property C in the classical dimension theory and compare it with the original.
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On Burghelea Conjecture
We prove the Burghelea Conjecture for groups satisfying some additional cohomological property.
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On some problems related to the Hilbert-Smith conjecture
We present a reduction of the Hilbert-Smith conjecture in the case of the finite dimensional orbit space to some algebraic topology problems.
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On homology of complements of compact sets in Hilbert Cube
We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of the complement results when (a) $X$ is weakly infinite dimensional; (b) $X$ has finite cohomological dimension.
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The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces
We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map for non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus $>3$ is four.
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On Gromov's conjecture for totally non-spin manifolds
Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.
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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 $q$.
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On Gromov's positive scalar curvature conjecture for virtual duality groups
We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $π_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.
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On topological complexity of twisted products
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(π_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
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On macroscopic dimension of universal coverings of closed manifolds
We give a homological characterization of $n$-manifolds whose universal covering $\Wi M$ has Gromov's macroscopic dimension $\dim_{mc}\Wi M n$.
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On cohomology of the Higson compactification of hyperbolic spaces
We show that in dimensions $>1$ the cohomology groups of the Higson compactification of the hyperbolic space $\H^n$ with respect to the $C_0$ coarse structure are trivial. Also we prove that the cohomology groups of the Higson compactification of $\H^n$ for the bounded coarse structure are trivial in all even dimensions.
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On dimensionally exotic maps
We call a value $y=f(x)$ of a map $f:X\to Y$ dimensionally regular if $\dim X\le \dim(Y\times f^{-1}(y))$. It was shown in \cite{first-exotic} that if a map $f:X\to Y$ between compact metric spaces does not have dimensionally regular values, then $X$ is a Boltyanskii compactum, i.e. a compactum satisfying the equality $\dim(X\times X)=2\dim X-1$. In this paper we prove that every Boltyanskii compactum $X$ of dimension $\dim X \geq 6$ admits a map $f:X\to Y$ without dimensionally regular values. Also we exhibit a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.
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On topological complexity and LS-category
We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample to the conjecture asserting that $TC(X')\le TC(X)$ for any covering map $p:X'\to X$. We discuss a possible reduction of the monoidal topological complexity to the LS-category. Also we apply the LS-category to give a short proof of the Arnold-Kuiper theorem.
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