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 X, 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 X. The abelian group of n-dimensional chains then have boundary functions or boundary maps (or boundary morphisms, since these chains form abelian groups) to the abelian group of n-1-dimensional chains.

The homology groups H_{n}(X) are then given as the quotient of the group of cycles (kernels of the n-th boundary morphisms) by the group of boundaries (images of the n+1-th boundary morphisms).

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

The cohomology groups H^{n}(X,G) are defined, dually to the homology groups, as the quotient of the the group of cocycles (kernels of the n-th coboundary morphisms) by the group of coboundaries (images of the n-1-th coboundary morphisms).

We now review some important concepts in homotopy theory. The n-th homotopy group \pi_{n}(X) of a space X is defined as the group [S^{n},X] of homotopy classes of functions from the n-dimensional sphere (also simply called the n-sphere) to X. equipped with appropriate basepoints. The homotopy group \pi_{1}(X) is called the fundamental group of X.

An Eilenberg-MacLane space, written K(G,n), for a certain group G and a certain natural number n, is a topological space characterized by the property that its n-th homotopy group is G 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 \mathbb{Z}, 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 K(\mathbb{Z},1).

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

\displaystyle \tilde{H}^{n}(X, G)\cong [X, K(G,n)]

where the symbol “\cong” denotes an isomorphism of groups and the \tilde{H}^{n}(X,G) are the reduced cohomology groups, obtained by “dualizing” the augmented chain complex of reduced homology, which has

\displaystyle ...\xrightarrow{\partial_{2}} C_{1}\xrightarrow{\partial_{1}} C_{0}\xrightarrow{\epsilon} \mathbb{Z}\rightarrow 0

instead of the usual

\displaystyle ...\xrightarrow{\partial_{2}} C_{1}\xrightarrow{\partial_{1}} C_{0}\xrightarrow{\partial_{0}} 0

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 H_{n}(X) and the reduced homology groups \tilde{H}_{n}(X):

\displaystyle H_{n}(X)=\tilde{H}_{n}(X)\oplus H_{n}(*)

Let X_{+} be the topological space obtained by adjoining a disjoint basepoint to X. Then

\displaystyle H_{n}(X)=\tilde{H}_{n}(X_{+})

Similarly, for cohomology, we have the following relations:

\displaystyle H^{n}(X)=\tilde{H}^{n}(X)\oplus H^{n}(*)

\displaystyle H^{n}(X)=\tilde{H}^{n}(X_{+})

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 X and Y with respective basepoints x_{0} and y_{0}, is the topological space, written X\wedge Y, given by the quotient space (X\coprod Y)/\sim under the identification x_{0}\sim y_{0}. One can think of the space X\wedge Y as being obtained from X and Y by gluing their basepoints together. For example, the wedge sum S^{1}\wedge S^{1} of two circles can be thought of as the “figure eight”.

The smash product of two topological spaces X and Y, again with respective basepoints x_{0} and y_{0}, is the topological space, written X\vee Y, given by the quotient space (X\times Y)/(X\wedge Y). What this means is that we take the cartesian product of X and Y, and then we collapse a copy of the wedge sum of X and Y containing the basepoint into the basepoint.

The suspension of a topological space X is the topological space, written SX, given by the quotient space (X\times I)/\sim under the identifications (x_{1},0)\sim (x_{2},0) and (x_{1},1)\sim (x_{2},1) for any x_{1}, x_{2}\in X, where I is the unit interval [0,1]. When X is the circle S^{1}, then X\times I is the cylinder, and SX is the cylinder with both ends collapsed into points. The space SX also looks like two cones with their bases glued together.

The reduced suspension of a topological space X with basepoint x_{0} is the topological space, written \Sigma X, given by the quotient space (X\times I)/(X\times \{0\}\cup X\times \{1\}\cup\{x_{0}\}\times I). This can be thought of as taking the suspension SX of X 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 S^{1}\vee X of the circle S^{1} and a topological space X is homeomorphic to the reduced suspension \Sigma X of X. Second, the smash product S^{m}\vee S^{n} of an m-dimensional sphere S^{m} and an n-dimensional sphere S^{n} is homeomorphic to an m+n-dimensional sphere S^{m+n}.

The loop space \Omega X of a space X with a chosen basepoint x_{0} is the topological space whose underlying set is the set of loops beginning and ending at  x_{0} 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:

\displaystyle [\Sigma X, Y]\cong [X, \Omega Y]

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

\Omega K(G,n)\cong K(G, n-1)

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

\Omega^{2} K(G,n)=\Omega (\Omega K(G,n))\cong K(G, n-2)

More generally, we will have

\Omega^{m} K(G,n)=\Omega K(G,n)\cong K(G, n-m) for m<n

We now define the concepts of prespectra and spectra. A prespectrum T is a sequence of spaces T_{n} with basepoints and basepoint-preserving maps \sigma: \Sigma T_{n}\rightarrow T_{n+1}. A spectrum E is a prespectrum with adjoints \tilde{\sigma}: E_{n}\rightarrow \Omega E_{n+1} 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

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One thought on “Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories

  1. Pingback: Varieties and Schemes Revisited | Theories and Theorems

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