Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories

In Homotopy Theory we explained the concept of homotopy and homotopy groups to study topological spaces and continuous functions from one topological space to another. Meanwhile, in Homology and Cohomology, we introduced the idea of homology and cohomology for the same purpose. In this post, we discuss one certain way in which homotopy and cohomology may be related to each other. Since homology and cohomology are related, being “duals” of each other in some sense, by extension this also relates homotopy and homology.

We recall first how we defined homology: Given a space , we construct a chain complex out of its subspaces. For subspaces of the same dimension, we must have the structure of an abelian group, so that these subspaces form chains on . The abelian group of -dimensional chains then have boundary functions or boundary maps (or boundary morphisms, since these chains form abelian groups) to the abelian group of -dimensional chains.

The homology groups  are then given as the quotient of the group of cycles (kernels of the -th boundary morphisms) by the group of boundaries (images of the -th boundary morphisms).

We also recall how we defined cohomology in terms of the homology. We define cochains on as the abelian group of the functions from chains on to some other abelian group . Together with the appropriate coboundary morphisms, they form what is called a cochain complex.

The cohomology groups  are defined, dually to the homology groups, as the quotient of the the group of cocycles (kernels of the -th coboundary morphisms) by the group of coboundaries (images of the -th coboundary morphisms).

We now review some important concepts in homotopy theory. The -th homotopy group of a space is defined as the group of homotopy classes of functions from the -dimensional sphere (also simply called the -sphere) to . equipped with appropriate basepoints. The homotopy group  is called the fundamental group of .

An Eilenberg-MacLane space, written , for a certain group and a certain natural number , is a topological space characterized by the property that its -th homotopy group is and all its other homotopy groups are trivial. In Category Theory we mentioned that the fundamental group of a circle is the group of integers , corresponding to the number of times a loop winds around the circle before returning to its basepoint, with the direction taken into account. All the other homotopy groups of the circle are actually trivial, which makes it an Eilenberg-Maclane space .

For certain kinds of topological spaces called CW-complexes (spaces which can be built out of “cells” which are topological spaces homeomorphic to -dimensional open or closed balls) we have the following interesting relationship between homotopy and cohomology:

where the symbol “” denotes an isomorphism of groups and the are the reduced cohomology groups, obtained by “dualizing” the augmented chain complex of reduced homology, which has

instead of the usual

for the purpose of making the homology groups of a topological space consisting of a single point trivial. We have the following relation between the homology groups and the reduced homology groups :

Let be the topological space obtained by adjoining a disjoint basepoint to . Then

Similarly, for cohomology, we have the following relations:

The idea of using the homotopy classes of functions from one space to another to define some kind of “generalized cohomology theory” leads to the theory of “spectra” in algebraic topology (it should be noted that the word “spectra” or “spectrum” has many different meanings in mathematics, and in this post we are referring specifically to the concept in algebraic topology).

We first introduce some definitions. The wedge sum of two topological spaces and with respective basepoints and , is the topological space, written , given by the quotient space under the identification . One can think of the space  as being obtained from and by gluing their basepoints together. For example, the wedge sum of two circles can be thought of as the “figure eight”.

The smash product of two topological spaces and , again with respective basepoints  and , is the topological space, written , given by the quotient space . What this means is that we take the cartesian product of and , and then we collapse a copy of the wedge sum of and containing the basepoint into the basepoint.

The suspension of a topological space is the topological space, written , given by the quotient space under the identifications and  for any , where is the unit interval . When is the circle , then is the cylinder, and  is the cylinder with both ends collapsed into points. The space  also looks like two cones with their bases glued together.

The reduced suspension of a topological space with basepoint is the topological space, written , given by the quotient space . This can be thought of as taking the suspension of and then collapsing a copy of the unit interval containing the basepoint into the basepoint.

We note a couple of results regarding smash products and reduced suspensions. First, the smash product of the circle and a topological space is homeomorphic to the reduced suspension of . Second, the smash product of an -dimensional sphere and an -dimensional sphere is homeomorphic to an -dimensional sphere .

The loop space of a space with a chosen basepoint is the topological space whose underlying set is the set of loops beginning and ending at and equipped with an appropriate topology called the compact-open topology.

We have the following relation between the concepts of reduced suspension and loop space:

By the definition of homotopy groups and of Eilenberg-MacLane spaces, together with the properties of the reduced suspension and of the smash product, this implies that

where the symbol in this context denotes homeomorphism of topological spaces. We can repeat the process of taking the loop space to obtain

More generally, we will have

for

We now define the concepts of prespectra and spectra. A prespectrum  is a sequence of spaces with basepoints and basepoint-preserving maps . A spectrum  is a prespectrum with adjoints which are homeomorphisms.

The Eilenberg-MacLane spaces form a spectrum, called the Eilenberg-MacLane spectrum. There are other spectra that will result in other generalized cohomology theories, which are defined to be functors (see Category Theory) from the category of pairs from pairs of topological spaces to the category of abelian groups, together with a natural transformation corresponding to a generalization of the boundary map, required to satisfy a set of conditions called the Eilenberg-Steenrod axioms. It follows from a certain theorem, called the Brown representability theorem, that every generalized cohomology theory comes from some spectrum, similar to how ordinary cohomology comes from the Eilenberg-MacLane spectrum.

References:

Eilenberg-MacLane Space on Wikipedia

Spectrum on Wikipedia

Eilenberg-Steenrod Axioms on Wikipedia

Algebraic Topology by Allen Hatcher

A Concise Course in Algebraic Topology by J. P. May