of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they...

obstruction to a homotopy equivalence f : X → Y {\displaystyle f\colon X\to Y} of finite CW-complexes being a simple homotopy equivalence is its Whitehead...

homotopy equivalence) if the inclusion maps M ↪ W and N ↪ W {\displaystyle M\hookrightarrow W\quad {\mbox{and}}\quad N\hookrightarrow W} are homotopy...

sequence. simple simple-homotopy equivalence A map ƒ:X→Y between finite simplicial complexes (e.g., manifolds) is a simple-homotopy equivalence if it is...

In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on...

into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes...

fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information...

extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence. Complexes that do not have a free face cannot be collapsible...

said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a homotopy type. There...

simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion. The torsion of a homotopy equivalence...

C++. All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions...

theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. A covering space of a CW complex...

directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental...

and which sends each morphism to its chain homotopy equivalence class. Since every chain homotopy equivalence is a quasi-isomorphism, Q {\displaystyle Q}...

topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological...

2π. Regular 3D immersions of the Klein bottle fall into three regular homotopy classes. The three are represented by: the "traditional" Klein bottle;...

functors give a Quillen equivalence of closed model categories inducing an equivalence |•|: Ho(sSet) ↔ Ho(Top) between the homotopy category for simplicial...

example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Let X be a topological space...

extended set of equivalences is also explored in homotopy type theory. Here, type theory is extended by the univalence axiom ("equivalence is equivalent...

adequate equivalence relation ∼ {\displaystyle \sim } . As was conjectured by Grothendieck and shown by Jannsen, this category is semi-simple if and only...

long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. The homotopy fiber has a simple description for a continuous...

connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers Q...

the inclusion M ↪ W {\displaystyle M\hookrightarrow W} is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion...

homotopy groups, S 0 {\displaystyle S^{0}} is the zero-sphere (i.e. two points) and [ U , W ] {\displaystyle [U,W]} denotes the homotopy equivalence of...

property defines an equivalence relation between chain maps. Let X and Y be topological spaces. In the case of singular homology, a homotopy between continuous...

the homotopy equivalences F, f 0 {\displaystyle f_{0}} and f 1 {\displaystyle f_{1}} to be simple homotopy equivalences then we obtain the simple structure...

In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It...

higher-order logic, and homotopy type theory. To define a quotient type, one typically provides a data type together with an equivalence relation on that type...

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely, two rings...

since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example...