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Concept
Spectrum (topology)
Summary
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory,there exist spaces such that evaluating the cohomology theory in degree on a space is equivalent to computing the homotopy classes of maps to the space , that is.Note there are several different of spectra leading to many technical difficulties, but they all determine the same , known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
There are many variations of the definition: in general, a spectrum is any sequence of pointed topological spaces or pointed simplicial sets together with the structure maps , where is the smash product. The smash product of a pointed space with a circle is homeomorphic to the reduced suspension of , denoted .
The following is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence of CW complexes together with inclusions of the suspension as a subcomplex of .
For other definitions, see symmetric spectrum and simplicial spectrum.
One of the most important invariants of spectra are the homotopy groups of the spectrum. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum define the homotopy group as the colimitwhere the maps are induced from the composition of the map (that is, given by functoriality of ) and the structure map . A spectrum is said to be connective if its are zero for negative k.
Eilenberg–Maclane spectrum
Consider singular cohomology with coefficients in an abelian group . For a CW complex , the group can be identified with the set of homotopy classes of maps from to , the Eilenberg–MacLane space with homotopy concentrated in degree . We write this asThen the corresponding spectrum has -th space ; it is called the Eilenberg–MacLane spectrum of .
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