arXiv research

Characterizes compact complex surfaces with finite homotopy rank-sum.

Characterizes Stein surfaces with finite homotopy rank-sum.

Study on descent properties of complex affine surfaces under proper morphisms.

New gossip algorithms improve robustness of rank-based statistics in decentralized systems.

Finite type and finitely generated homotopy groups for manifold automorphisms.

Classifies compact spaces by shape, finite spaces by weak homotopy.

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…

4-manifolds with specific groups have unique homotopy types.

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

This paper categorifies Quinn's TQFTs and computes them for specific omega-groupoids.

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

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy ty…

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

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial …

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of…

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

Embeddings of mapping tori for end-periodic graph maps are proven.

The paper explores conditions for compactness and finiteness in stratified homotopy theory.

New invariant detects non-homotopy equivalent 4-manifolds.

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

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regu…

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.

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…

We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.

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.

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

If a finite group $G$ is isomorphic to a subgroup of $SO(3)$, then $G$ has the D2-property. Let $X$ be a finite complex satisfying Wall's D2-conditions. If $π_1(X)=G$ is finite, and $χ(X) \geq 1-Def(G)$, then $X \vee S^2$ is simple homotopy equivalent to a finite $2$-complex, whose simple homotopy type depends only on …

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…

Simplified proofs for splitting homotopy idempotents.

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-…

Locally flat submanifolds have finite CW complex complements.

Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…

Complex equivalence classes found in graph homotopy.

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…

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 $K(π_{1}(X),1)$.

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.

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

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

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

The study explores ends in coarse homotopy of proper geodesic spaces.

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

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…

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…

Uniform entropy bound for Ricci shrinkers with bounded curvature.

Stable approach solves equivariant Hopf theorem for G-manifolds.

We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.

This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a stron…