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
Eilenberg-Steenrod Axioms on Wikipedia
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