arXiv research

New examples of manifolds that are homotopy but not simple homotopy equivalent.

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

Proves loop coproduct invariance under simple homotopy equivalences.

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

Determines regular homotopy classes for link immersions of simple singularities.

We prove that the higher signature for any close oriented manifold is a simple-homotopy invariant.

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 …

RSHT algorithm simplifies complex shapes to points.

Homotopy types of curve and arc complexes are studied.

The paper studies obstructions to homotopy invariance of loop coproducts.

Finite type and finitely generated homotopy groups for manifold automorphisms.

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…

Simple construction of Rumin algebra for contact manifolds.

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…

Simplified proofs for splitting homotopy idempotents.

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 on the parity of fold map singular points, showing non-invariance for odd-dimensional manifolds.

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem…

The study connects hypergraphs to strong homotopy Lie algebras.

Study shows moduli space of fibrations has specific homotopy types.

Study homotopy motions of surfaces in 3-manifolds.

Inverse function theorem and homotopy description for L-infinity bundles.

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 …

In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.

We show that any 3-dimensional homotopy lens space M^3 that is simple-homotopy equivalent to a lens space L(p,q) is topologically s-cobordant to the lens space. It follows that M has the same multi-signature as L(p,q) and the action of π_1(M) on the universal cover of M embeds in an orthogonal action on S^7.

New algebra models refine complex manifold homotopy groups.

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable …

Polynomial maps are shown to be Serre fibrations under specific conditions.

Geometric proof of curve characterization using loop-bundles.

Study the space of embeddings of split links in 3D and 4D.

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted th…

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singu…

Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.

The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connect…

We prove that if $Γ$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $Γ$-equivariantly homotopy equivalent to a proper cocompact $Γ$-CW complex of dimension the virtual cohomological dimension of $Γ$.

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for an…

Given two compact n-dimensional manifolds in the smooth, piecewise linear or topological categories, basic results of B. Mazur and others give simple criteria for determining whether their products with Euclidean spaces of sufficiently large dimension are isomorphic in the given category. This paper studies such questi…

Connected graph for twice-punctured torus curves.

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surfa…

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with $n$ self-crossings requires $Θ(n^{3/2})$ homotopy moves in the worst case. Our algorithm improves the best previou…

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when the later are given their natural CW-decomposition with unique cells of order 0, $(n-1)$ and $n$; a result resembling J.H.C. Whi…

Paper explores relationships between triple chords and a specific homotopy relation in knot theory.

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…

The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk …

Paper generalizes wrinkled embedding concept to jet spaces.

We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar constru…