The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.
problem Proving the Singer conjecture and refining Gromov's inequality for aspherical complex surfaces.
method Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group, and refinement of Gromov's inequality for non-residually finite surfaces.
result Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group.
Researchers propose a new approach to the Hopf problem for aspherical varieties.
problem The Hopf problem for aspherical smooth projective varieties.
method An approach recently proposed by Liu, Maxim, and Wang in [LMW21].
result An intriguing conjecture regarding the geography of aspherical surfaces of general type.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.
Study new bounds on TC of spaces with subgroup inclusions.
problem Lower bounds on TC of spaces with subgroup inclusions.
method Generalizes TC results from aspherical spaces to spaces with subgroup inclusions.
result Establishes new lower bounds on sequential TCs of aspherical spaces.
Study shows some complex shapes don't fit a certain property.
problem Some aspherical manifolds lack the Bounded Index Property (BIP).
method Analyzes specific aspherical manifolds to prove they don't have BIP.
result Proves existence of aspherical manifolds without BIP.
New method controls surface extrinsic diameter for positive scalar curvature metrics.
problem Preventing complete metrics with positive scalar curvature on surfaces within manifolds.
method Interior control for extrinsic diameter of surfaces with positive scalar curvature.
result Closed aspherical manifolds cannot have complete metrics with positive scalar curvature when subsets are removed.
The paper describes the structure of injective LOT-complexes and proves they are aspherical.
problem The unresolved asphericity question for labeled oriented trees encoding spines of ribbon discs.
method Complete description of the link of a reduced injective LOT complex, proving asphericity.
result Reduced injective LOT complexes are aspherical, with specific conditions for non-boundary sub-LOTs.
The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …
An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open alo…
Diagrammatic reducibility DR and its generalization vertex asphericity VA are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes (L,K), where K is a subcomplex of L. We show that a relative weight test holds…
We show that for each aspherical compact complex surface X whose fundamental group π fits into a short exact sequence 1→K→π→π1(S)→1 where S is a compact hyperbolic Riemann surface and the group K is finitely-presentable, there is a complex structure on S and a nonsingular holomorphic fibr…
The study shows that certain cubical presentations lead to aspherical spaces.
problem Understanding the asphericity of cubical presentations in 2D.
method Analyzing the second homotopy group of coned-off spaces associated with cubical presentations.
result The coned-off space is aspherical under specific conditions.
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …
Study pro-p completions of orientable PD_n groups, proving best results in three cases.
problem Understanding pro-p completions of orientable PD_n groups. method Examined four cases of orientable PD_3-groups and some PD_n groups (n≤5), providing examples and proving best results in three cases.
result Best results in three out of four cases of orientable PD_3-groups.
Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
In this paper we show that for a large class C of 4-manifolds each member of C has only finitely many simple homotopy type upto s-cobordism. This result generalizes a similar result of Hillman for certain complex surfaces. We also present a correction in the proof of Hillman's result.
Study symplectically aspherical Kähler manifolds with unique properties.
problem Existence and properties of symplectically aspherical Kähler manifolds.
method Detailed study and analysis of geometric and topological features.
result Existence of symplectically aspherical Kähler manifolds with large fundamental groups.
We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical Kähler manifold with a cu…
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …
We describe the second homotopy group of any CW-complex K by analyzing the universal cover of a locally finite model of K using the notion of G-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-…
The paper improves bounds on topological complexity for certain manifolds.
problem Determining bounds on topological complexity for specific manifolds.
method Analyzing cohomology classes and their pullbacks, using Gromov norm.
result Improved bounds on topological complexity for connected sums.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
A labeled oriented graph (LOG) is an oriented graph with a labeling function from the edge set into the vertex set. The complexity of a LOG is the minimal cardinality of an initial set S of vertices such that every vertex can be reached successively from S only using edges with labels in S or already visited vert…
Study irrational pencils on complex manifolds, finding non-finitely generated homology.
problem Understanding the homology of the kernel induced by irrational pencils on complex manifolds.
method Analyzing critical points and homology of fundamental groups of complex manifolds.
result Homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated.
If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nont…
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known…
Let X be a finite aspherical CW-complex whose fundamental group π1(X) possesses a subnormal series π1(X)⊳Gm⊳...⊳G0 with a non-trivial elementary amenable group G0. We investigate the L2-invariants of the universal covering of such a CW-complex X. We show that the Novikov-Shubin invarian…
We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete…
Surveying topological complexity of graph configurations, unifying traditional and modern approaches.
problem Understanding the topological complexity of configuration spaces of graphs.
method Exploring traditional cohomology methods and modern asphericity/fundamental group approaches.
result Unified understanding of topological complexity through both traditional and modern methods.
The paper explores automorphism groups of parabolic structures on aspherical manifolds.
problem Characterizing the automorphism groups of parabolic structures on aspherical manifolds.
method Analyzing properties of closed aspherical parabolic ${\sfG}$-manifolds and their automorphism groups.
result Certain parabolic ${\sfG}$-structures impose strong restrictions on the topology of compact aspherical manifolds.
We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …
Proves connection between ℓ2-Betti numbers and BNSR invariants.
problem Relationship between ℓ2-Betti numbers and BNSR invariants. method Analyzes ℓ2-Betti numbers and BNSR invariants of groups. result If the nth ℓ2-Betti number is non-zero, then the nth BNSR invariant over Q is empty. Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …
Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.
problem Characterize minimal volume entropy for aspherical simplicial complexes with these groups as fundamental groups.
method Algebraic and geometric characterization, using fiber π1-growth collapse and non-collapsing assumptions. result Provide bounds and criteria for minimal volume entropy in aspherical simplicial complexes.
Aspherical configuration Lie groupoids complement is proven for a class of orbifolds.
problem Proving aspherical property of configuration Lie groupoids complement for orbifolds.
method Generalization to Lie groupoids, focusing on orbifolds.
result Complement is aspherical for a class of aspherical 2-dimensional orbifolds.
We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fund…
We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes P-adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove…
The paper shows geometric realisation over specific groups and knots.
problem Geometric realisation of modules over aspherical groups and knots.
method Using extensions of scalars of relation modules and constructing specific 2-complexes.
result Exotic presentations of groups and stably free non-free modules over Baumslag-Solitar groups.
The paper shows how certain Kähler groups are uniquely determined by their profinite completions.
problem Understanding the uniqueness of Kähler groups within residually finite groups.
method Holomorphic fibrations and profinite completions of fundamental groups.
result Aspherical smooth projective varieties are determined by their algebraic fundamental groups.
New groups with special properties found.
problem Finding new groups with specific geometric properties.
method Proved actions on CAT(0) cubical complexes under certain conditions.
result Many groups admit cocompact actions on CAT(0) cubical complexes.
The paper proves spaces associated to certain cubical presentations are aspherical.
problem Proving the asphericity of cubical presentations.
method Using coned-off spaces and properties of fundamental groups.
result The fundamental groups of cubical presentations have aspherical associated spaces.
We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings…