Stable homotopy category

Stable homotopy category

In homotopy theory, the stable homotopy category can be thought to be related to the category of spaces and continuous maps in the same way that stable homotopy groups are related to (standard) homotopy groups.

Formally, the objects in the category are omega spectra, and the morphisms are homotopy classes of maps of spectra; hence, we may think of this category as eliminating the distinction between spaces X and Y which are not homeomorphic, but satisy

:Sigma^n X = Sigma^n Y

for some natural number n. (Here Sigma denotes the suspension functor.)


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Stable homotopy theory — In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the… …   Wikipedia

  • Spectrum (homotopy theory) — In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, all of which give the same homotopy category.Suppose we… …   Wikipedia

  • Triangulated category — A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t category is a triangulated category with a t… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Michael J. Hopkins — Mike Hopkins in Oberwolfach 2009 Born April 18, 1958 …   Wikipedia

  • Freudenthal suspension theorem — In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the… …   Wikipedia

  • Mapping class group — In mathematics, in the sub field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of symmetries of the space. Contents 1 Motivation 2… …   Wikipedia

  • Spectral sequence — In the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a… …   Wikipedia

  • Cobordism — A cobordism (W;M,N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint… …   Wikipedia

  • Segal conjecture — Segal s Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group G to the stable cohomotopy of the classifying space BG .… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”