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

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

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...

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...

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

every homotopy equivalence is an equivalence (in Voevodsky's good coherent sense), based on the idea from category theory of improving equivalences to adjoint...

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

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...

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...

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

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

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....

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

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...

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...

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...

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

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...

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

particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations'...

connected spaces and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected loop spaces. The homotopy fiber of any map between...

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

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

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

fibre S1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B...

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...

in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...