Eilenberg-Mac Lane spectrum
Special and general types
Stable Homotopy theory
For an abelian group, the Eilenberg-Mac Lane spectrum is the spectrum that represents the ordinary cohomology theory/ordinary homology with coefficients in .
By default is taken to be the integers and hence “the Eilenberg-MacLane spectrum” is , representing integral cohomology.
As a symmetric spectrum: (Schwede 12, example I.2.6)
As symmetric monoidal -groupoids
As a symmetric monoidal ∞-groupoid the Eilenberg–Mac Lane spectrum is the abelian group of integers (under addition)
Here the set is regarded as a discrete groupoid (one object per integer, no nontrivial morphisms) whose symmetric monoidal structure is that given by the additive group structure on the integers.
Accordingly, the infinite tower of suspensions induced by this is the sequence of ∞-groupoids
that in this case happen to be strict omega-groupoids. The strict omega-groupoid has only identity -morphisms for all , except for , where are the endomorphisms of the unique identity -morphism.
The strict ∞-groupoid is the one given under the Dold-Kan correspondence by the crossed complex of groupoids that is trivial everywhere and has the group in degree .
Under the Quillen equivalence
between infinity-groupoids and topological spaces (see homotopy hypothesis) this sequence of suspensions of maps to the sequence of Eilenberg–Mac Lane spaces
that give the Eilenberg–Mac Lane spectrum
chromatic homotopy theory
Ordinary homology spectra split
ordinary homology spectra split: For any spectrum and an Eilenberg-MacLane spectrum, then the smash product (the -ordinary homology spectrum) is non-canonically equivalent to a product of EM-spectra (hence a wedge sum of EM-spectra in the finite case).
Textbook accounts include
Lecture notes include
Revised on April 15, 2016 09:58:32
by Urs Schreiber