We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A-homotopy type. Arithmetic considerations involving the sheaf of A-connected components lead us to introduce several di...

We determine the homological residue fields, in sense of tensor-triangular geometry, a series concrete examples ranging from topological stable homotopy theory to modular representation finite groups.

Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of d...

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 connected sums of certain geometric 4-manifolds. Most of these homotopy fibers h...

in this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  we relate the cohomology of a normal subgroup $n$ of a topological group $g$ and the quotient $g/n$ to the cohomology of $g$. we get the inflation-restriction exact sequence. also, we obtain a seven-term exact cohomology sequence up to dimension 2. we give an interpretation of the first non-a...

I start by telling a bit about stable homotopy theory, to give a flavor of the world we’ll be working in today. One of the key ideas in algebraic topology is to use algebraic invariants to distinguish topological spaces. This leads to H∗, H ∗, and π∗. The first two are relatively easy to compute, but they are coarse invariants: they don’t capture enough of the information of the space. The last...

Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. This technical result allows an extension of Cappell’s 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the splitting and fibering problems for certain 5-manifolds mapping to the circle. For ex...

In this paper, we aim to develop a suitable homotopy theory of finite topological graphs by Khalimsky arcs. The notions developed are considered for the investigation of the algebraic invariants of topological graphs for their topological and graphical classifications. Keywords—Connected ordered topological spaces, digital topology, homotopy, Khalimsky spaces, topological graphs.

Using operad methods and functional homology operations, we obtain inductive formulae for the differentials of the Adams spectral sequence of stable homotopy groups of spheres. The Adams spectral sequence was invented by Adams [1] almost fifty years ago for the calculation of stable homotopy groups of topological spaces (in particular, those of spheres). The calculation of the differentials of ...

Topological cospans and their concatenation, by pushout, appear in the theories of tangles, ribbons, cobordisms, etc. Various algebraic invariants have been introduced for their study, which it would be interesting to link with the standard tools of Algebraic Topology, (co)homotopy and (co)homology functors. Here we introduce collarable (and collared) cospans between topological spaces. They ge...