arXiv research

Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.

Characterizes compact complex surfaces with finite homotopy rank-sum.

Characterizes Stein surfaces with finite homotopy rank-sum.

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-…

Uniform entropy bound for Ricci shrinkers with bounded curvature.

Generalizes $π_2$-diffeomorphism finiteness to non-zero first homotopy groups.

The paper shows geometric realisation over specific groups and knots.

We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds.

Finite type and finitely generated homotopy groups for manifold automorphisms.

The paper defines and proves the existence of train track maps on graphs of groups.

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring po…

We prove the vanishing of higher A-hat-genera, in the sense of Browder and Hsiang, on smooth manifolds with effective circle actions and with finite second and fourth homotopy groups

In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 1. For any given numbers $m$, $C$ and $D$, the class of $m$-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature $\vert …

The study shows that certain manifolds with positive curvature cannot contain specific geometric structures.

4-manifolds with specific groups have unique homotopy types.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

We show that open 3-manifolds that have a locally finite decomposition along 2-spheres are characterized by the existence of a Riemannian metric with respect to which the second homotopy group of the manifold is generated by small elements.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central…

Survey on finite group actions on CW-complexes homotopy to spheres.

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

Derives geometrically a description of a 3-manifold's second homotopy group.

We compute the first and second homotopy groups of a class of contact toric manifolds in terms of the images of the associated moment map.

Computes homotopy groups of embedding spaces of arcs or circles in 4-manifolds.

Constructing manifold bundles from orbifolds and proving the existence of free subgroups in second homotopy groups.

Some properties of [L]-homotopy group for finite complex L are investigated. It is proved that for complex L whose extension type lying between Sn and Sn+1 n-th [L]-homotopy group of Sn is isomorphic to Z.

New invariant detects non-homotopy equivalent 4-manifolds.

We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold. We characterize when th…

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of th…

Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.

Homotopy types of 4-manifolds tied to their fundamental groups.

We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was establi…

Let X be a finite CW-complex of dimension q. If its fundamental group $π_{1}(X)$ is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups $π_{i}(X)$ is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space $K(π_{1}(X),1)$.

Groups of homotopy equivalences of graphs help realize compact subgroups.

We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…

Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.

Smooth actions of infinite groups linked to homotopy theory.

Simplified proofs for splitting homotopy idempotents.

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ whose universal cover $\til…

Researchers show a complex structure is not a counterexample to a topological problem.

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was prove…

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite con…

New examples of manifolds with similar homotopy but different simple homotopy types.

The paper classifies links up to link-homotopy using claspers.

We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply-connected four manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.

Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.

Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calcula…