Spectrum (topology)

In the mathematical subfield of algebraic topology spectra are used to define generalized homology theories.

  • 4.1 Example
  • 5.1 Examples
  • 5.2 Calculation
  • 5.3 Brownian Darstellbarkeitssatz

Definition

A spectrum is a consequence dotted spaces dotted with continuous maps

Herein, the reduced device to attach from.

Because the reduced device to attach is linksadjungiert to form the loop space corresponds to a unique continuous map up to homotopy. A spectrum is a spectrum if the pictures are homeomorphisms.

One finds in the literature, other definitions. For example, the above-defined spectra as Präspektrum and the spectra are then referred to as a spectrum. These names can be assigned to any Präspektrum by a spectrum, its Spektrifizierung.

A morphine between spectra and is a family of continuous maps with for all.

Examples

  • Einhängungsspektren: For a topological space together with the canonical homeomorphisms a spectrum. It is referred to as Einhängungsspektrum the room. General shape of the spectra are referred to as Einhängungsspektren, being meant for a spectrum with the spectrum.
  • Spheres Spectrum: Einhängungsspektrum -dimensional sphere is the spectrum of spheres, and is denoted by. In this case, therefore, and the canonical homeomorphism.
  • Eilenberg - MacLane spectrum: For an abelian group, the Eilenberg - MacLane spaces form with a spectrum and given by the set of Whitehead homotopy equivalence. This range is also referred to.
  • Thom spectrum: The Thom space of the universal vector bundle over the Grassmann manifolds form a spectrum. The texture mapping is in this case induced by the classifying of the image vector bundle Figure
  • Topological K-theory spectrum: This spectrum is defined by for all, with the ascending union of unitary groups and their classifying space.
  • Spectra: Be an infinite loop space, then defines a spectrum.
  • Algebraic K-theory spectrum: For a commutative ring with unity, the use of the plus - construction on the classifying space of an infinite loop space and therefore defines a spectrum.

Homotopy groups of spectra

The kth homotopy a spectrum defined by

The homotopy groups of a Einhängungsspektrums are called stable homotopy groups by:

For spectra is already.

Examples

  • The stable homotopy groups of spheres are the homotopy groups of the sphere spectrum.
  • The algebraic K-theory of a commutative ring with unity is obtained for by definition of homotopy algebraic K-theory spectrum.
  • The Kobordismusgruppe unoriented manifolds is isomorphic to the -th homotopy group of the Thom spectrum.

Equivalences

For morphisms of spectra, the following analogue of the set of Whitehead applies:

A morphism of spectra induces an isomorphism of all homotopy groups if and only if the induced morphism in the homotopy category of spectra is an isomorphism. Such maps are called equivalences.

Generalized homology theories

A spectrum defines a (reduced ) generalized homology theory by

Where the designated aid to the smash product by defined range.

Is particular.

Example

Is isomorphic to Kobordismusgruppe singular manifolds in.

Generalized cohomology theories

Each spectrum defines a generalized (reduced ) cohomology theory by

For topological spaces, the homotopy classes of continuous maps dotted called. ( It is said that the cohomology theory is represented by the spectrum. )

The corresponding unreduced cohomology theory is denoted by.

Examples

The Eilenberg - MacLane spectrum defines the singular homology, the topological K-theory spectrum defined topological K- theory.

Calculation

Generalized cohomology groups of a space can often be computed using the Atiyah - Hirzebruch spectral sequence. This is a spectral sequence converging to with -Term

With singular cohomology with coefficient group called.

Brownian Darstellbarkeitssatz

It follows from the Brownian Darstellbarkeitssatz that every generalized cohomology theory can be represented by a reduced spectrum represent.

Smash product

On a range and a space is defined by the range and the texture maps.

There is a going back to Adam's design, which allocates two spectra and a smash product, which has the following properties:

  • The smash product is a covariant functor of both arguments.
  • There are natural equivalences.
  • For each spectrum, and every CW complex there is a natural equivalence. In particular, for all CW complexes.
  • If an equivalence is, then.
  • For a family of spectra is an equivalence.
  • If a Kofaserung of spectra is then also.

Ring spectra

A ring spectrum is a spectrum with a smash product and with morphisms

The conditions

Suffice.

Mathematics Eilenberg-MacLane space Frank Adams
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