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...rt description|Offers a substitute for the absence of excision in homotopy theory}} ...substitute for the absence of [[Excision theorem|excision]] in [[homotopy theory]]. More precisely, let <math>(X; A, B)</math> be an [[excisive triad]] with ...

...escription|Two theorems needed for Quillen's Q-construction in algebraic K-theory}} ...eorems play central roles in Quillen's [[Q-construction]] in [[algebraic K-theory]] and are named after [[Daniel Quillen]]. ...

...motopy group|homotopy groups]] of a [[ring spectrum]] to be [[nilpotent]], in terms of the [[complex cobordism]] spectrum <math>\mathrm{MU}</math>. More ...=Goro Nishida|year=1973}} showed that elements of positive degree of the [[homotopy groups of spheres]] are nilpotent. This is a special case of the nilpotence ...

{{short description|Results on triad homotopy groups}} ...y | first2=William S. | authorlink2=William S. Massey| year=1949|title=The homotopy groups of a triad| ...

{{DISPLAYTITLE:Basic theorems in algebraic ''K''-theory}} {{short description|Four mathematical theorems}} ...

...topy Limits, Completions and Localizations|volume=304|series=Lecture Notes in Mathematics|first1=Aldridge K.|last1=Bousfield|author1-link=Aldridge Bousfi * <math>\pi</math> acts nilpotently<ref name=Dror71/> on the [[higher homotopy groups]] <math>\pi_i (X), i \ge 2</math>, i.e., there is a [[central series ...

...ellular on a subcomplex ''A'' of ''X'', then we can furthermore choose the homotopy to be stationary on ''A''. From an algebraic topological viewpoint, any map ...X'' can thus be connected to a 0-cell of ''Y'' by a path, but this gives a homotopy from ''f'' to a map which is cellular on the 0-skeleton of X. ...

...ort description|Mathematical result about equivariant K-theory in homotopy theory}} ...orem]] in [[mathematics]] about [[equivariant]] [[K-theory]] in [[homotopy theory]]. Let ''G'' be a [[compact group|compact]] [[Lie group]] and let ''X'' be ...

...)|weak equivalence]]. One of the main applications of quasifibrations lies in proving the [[Dold-Thom theorem]]. ...a quasifibration ''p'': ''E'' → ''B'' induces a [[long exact sequence]] of homotopy groups ...

...rson cohomology|BP-homology]], and are useful objects in [[stable homotopy theory]]. ...nted the way towards the [[nilpotence theorem|nilpotence]] and periodicity theorems.<ref>{{Cite book|url=https://books.google.com/books?id=xoM5DxQZihQC&q=Larry ...

...ategory]]. The study of such generalizations is known as [[higher category theory]]. ...ms have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by {{harvs|txt|authorlink=Jacob Luri ...

In [[homotopy theory]], a branch of [[mathematics]], the '''Barratt–Priddy theorem''' (also refe ...mpact-open topology]]). These maps are required to fix a basepoint <math>x\in S^n</math>, satisfying <math>f(x)=x</math>, and to have [[degree of a conti ...

...logy]] about the topology of the space of [[Flat convergence|flat cycles]] in a Riemannian manifold. ...Riemannian manifold Almgren isomorphism theorem implies that it is weakly homotopy equivalent to ...

...about the [[homotopy type]] of a topological space. In terms of [[category theory]], the fundamental groupoid is a certain [[functor]] from the category of t ...on, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indis ...

...group, and a [[chain homotopy]] corresponds to a [[simplicial homotopy]]. (In fact, the correspondence preserves the respective standard [[model category ...lisher=[[European Mathematical Society]] | location=Zurich | series=Tracts in Mathematics | isbn= 978-3-03719-083-8 | year=2011 | volume=15}}</ref> has a ...

| caption = Ravenel in 1978 | awards = [[Oswald Veblen Prize in Geometry|Veblen Prize]] (2022) ...

In [[mathematics]], the '''pseudoisotopy theorem''' is a theorem of [[Jean Cer ...fnof;'' preserves the level-sets <math>M \times \{t\}</math> for <math> t \in [0,1]</math>. ...

...can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law. ...[[Flat module|flat]] over <math>MU_*</math>, but that would be too strong in practice. Peter Landweber found another criterion: ...

...herent sheaves on a stack: this is one of the first motivations behind the theory. ...chester.edu/people/faculty/doug/mu.html|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1 ...

{{short description|On the homotopy groups of the infinite symmetric product of a connected CW complex}} ...used in doing so are [[quasifibration]]s. The theorem has been generalised in various ways, for example by the [[Almgren isomorphism theorem]]. ...