On totally c-symplectically aspherical manifolds
We construct examples of totally c-symplectically aspherical near-symplectic manifolds with non-trivial second homotopy group.
math.SG↗
arXiv subjects
Alexander Dranishnikov
Publications and source records attributed to Alexander Dranishnikov.
Symplectically aspherical K\"ahler manifolds, scalar curvature, and the fundamental group
We present a detailed study of closed smooth manifolds having K\"ahler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical K\"ahler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group \'a la Koll\'ar and the absence of K\"ahler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov--Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study K\"ahler cones on symplectically aspherical K\"ahler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.
math.GT↗
On distributional topological complexity of groups and manifolds
We prove the equality $\dTC(\Gamma)=\TC(\Gamma)$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. For the distributional topological complexity of lens spaces we prove the inequality $\dTC(L^n_p)\le 2p-1$ and for the distributional LS-category the inequality $d\cat(L^n_p)\le p-1$ which turns into equality for prime $p$ and $n>p$. We use these inequalities to bring counter-examples to the product formula for $d\cat$ and $\dTC$.
math.GT↗
Rudyak's conjecture for lower dimensional 1-connected manifolds
Rudyak's conjecture states that for any degree one map $f:M\to N$ between oriented closed manifolds there is the inequality $\cat (M)\ge \cat(N)$ for the Lusternik-Shnirelmann category. We prove the Rudyak's conjecture for $ n$-dimensional simply connected spin manifolds for $n\le 8$.
math.AT↗
A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shift
We construct an example announced in the title. It answers in a strong way a well-known open problem in topological dynamics. In fact our construction is an existence theorem. It is based on a Borsuk-Ulam type theorem whose proof heavily relies on p-adic completions of G-complexes and the equivariant Sullivan conjecture.
math.DS↗
Curvature, macroscopic dimensions, and symmetric products of surfaces
We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, we prove that symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and the second-named author. As a natural generalization of this circle of ideas, we address the Gromov--Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.
math.GT↗
Distributional Topological Complexity of groups
We study numerical invariants $d\TC(\Gamma)$ and $d\cat(\Gamma)$ of groups recently introduced in \cite{DJ} and independently in \cite{KW}. We compute $d\TC$ for finite cyclic groups $\mathbb Z_p$ with prime $p$ as well as for nonorientable surfaces of genus $g>3$ (for orientable surfaces it was computed in \cite{DJ}). We prove the formula $$d\TC(G\ast H)=\max\{d\TC (G),d\TC (H), \cd(G\times H)\}$$ for torsion free groups.
math.GT↗
Distributional Topological Complexity and LS-category
We define a new version of Topological Complexity (TC) of a space, denoted as $\text{dTC}$, which, we think, fits better for motion planning for some autonomous systems. Like Topological complexity, \text{dTC} is also a homotopy invariant. Also, $\text{dTC}$ has a corresponding analog, denoted as $\text{dcat}$, to the Lusternik-Schnirelmann category (cat). In this paper, we do computations and estimates for both $\text{dTC(X)}$ and $\text{dcat(X)}$ for some spaces $X$ as well as a comparison with $\text{TC(X)}$ and $\text{cat(X)}$.
math.GT↗
On Lipschitz cohomology of aspherical manifolds
We introduce the notion of Lipschitz cohomology classes of a group with local coefficients and reduce the Novikov higher signature conjecture for a group $\Gamma$ to the question whether the Berstein-Schwarz class $\beta_\Gamma\in H^1(\Gamma,I(\Gamma))$ is Lipschitz.
math.GT↗
On Gromov's conjecture for right-angled Artin groups
We prove that the universal covering $\Wi M$ of an $n$-dimensional closed spin PSC manifold $M$ whose fundamental group $\pi_1(M)$ is a right-angled Artin group has macroscopic dimension $\dim_{mc}\Wi M\le n-2$. This confirms Gromov's conjecture in the case of right-angled Artin groups.
math.GT↗
On cohomological dimension of group homomorphisms
The (co)homological dimension of homomorphism $\phi:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $\phi:G\to H$ of a geometrically finite group $G$ the homological dimension of $\phi$ equals the cohomological dimension, $hd(\phi)= cd(\phi)$. THEOREM 2. For every homomorphism $\phi:G\to H$ of geometrically finite groups $cd(\phi\times\phi)=2cd(\phi)$.
math.AT↗
Embedding of Higson compactification into the product of adelic solenoids
The Higson compactification of any simply connected proper geodesic metric space admits an embedding into a product of adelic solenoids that induces an isomorphism of 1-dimensional cohomology.
math.GN↗
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.
math.GR↗
On the LS-category of homomorphisms
We prove the equality $\cat(ϕ)=\cd(ϕ)$ for homomorphisms $ϕ:Γ\to Λ$ of a torsion free finitely generated nilpotent groups $Γ$ to an arbitrary group $Λ$. We construct an epimorphism $ψ:G\to H$ between geometrically finite groups with $\cat(ψ)> \cd(ψ)$.
math.AT↗
On Iwase's 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 Ganea's conjecture: ${\rm cat_{LS}} (M\times S^n)={\rm cat_{LS}} M+1$. 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: ${\rm cat_{LS}}(M\times M)=2{\rm cat_{LS}} M$. Also, we construct a map of degree one $$f:N\to M_2\times M_3$$ which reduces Rudyak's conjecture to the question whether ${\rm cat_{LS}}(M_2\times M_3)\ge 5$. We show that ${\rm cat_{LS}}(M_2\times M_3)\ge 4$.
math.AT↗
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 $u:M'\to K(π_1(M'),1)$ 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.
math.DG↗
Surgery Approach to Rudyak's 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, $r\ge 1$. If $N$ satisfies the inequality $\dim N \leq 2r \cat N - 3$, then for the Lusternik-Schnirelmann category $\cat M \geq \cat N$ .
math.AT↗
The Lusternik-Schnirelmann category of connected sum
We use the Berstein-Hilton invariant to prove the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for the Lustrnik-Schnirelmann category of the connected sum of closed manifolds $M_1$ and $M_2$.
math.AT↗