Homotopy excision theoremIn algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let be an excisive triad with nonempty, and suppose the pair is ()-connected, , and the pair is ()-connected, . Then the map induced by the inclusion , is bijective for and is surjective for . A geometric proof is given in a book by Tammo tom Dieck. This result should also be seen as a consequence of the most general form of the Blakers–Massey theorem, which deals with the non-simply-connected case.
Classification theoremIn mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following. The equivalence problem is "given two objects, determine if they are equivalent". A complete set of invariants, together with which invariants are solves the classification problem, and is often a step in solving it.
Hans FreudenthalHans Freudenthal ( – ) était un mathématicien juif allemand, naturalisé néerlandais, spécialiste en topologie algébrique mais dont les contributions ont largement débordé ce domaine. Il s'intéressa à l'enseignement des mathématiques. Il fut président de l'ICMI (Commission internationale de l'enseignement mathématique) et une récompense portant son nom est attribuée. On lui doit notamment le problème de Freudenthal, dans lequel « savoir que quelqu'un ne sait pas permet de savoir ».
Abstract simplicial complexIn combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
Ambient isotopyIn the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let and be manifolds and and be embeddings of in .
Théorème de WhiteheadEn théorie de l'homotopie (une branche des mathématiques et plus précisément de la topologie algébrique), le théorème de Whitehead établit que si une application continue f entre deux espaces topologiques connexes X et Y induit un isomorphisme sur tous leurs groupes d'homotopie, alors f est une équivalence d'homotopie dès que X et Y ont le type d'homotopie de CW-complexes. Ce résultat a été démontré par J. H. C. Whitehead dans deux articles de référence de 1949 et justifie l'introduction de la notion de CW-complexes faite dans ces articles.
