Eilenberg–MacLane space

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In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space[note 1] is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. If then G must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just .

A generalised Eilenberg–Maclane space is a space which has the homotopy type of a product of Eilenberg–Maclane spaces .

Examples[edit]

  • The unit circle is a .
  • The infinite-dimensional complex projective space is a model of . Its cohomology ring is , namely the free polynomial ring on a single 2-dimensional generator in degree 2. The generator can be represented in de Rham cohomology by the Fubini–Study 2-form. An application of is described as abstract nonsense.
  • The infinite-dimensional real projective space is a .
  • The wedge sum of k unit circles is a , where is the free group on k generators.
  • The complement to any knot in a 3-dimensional sphere is of type ; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.[1]
  • Any compact, connected, non-positively curved manifold M is a , where is the fundamental group of M.
  • An infinite lens space given by the quotient is a . This can be shown using the long exact sequence on homotopy groups for the fibration since because the infinite sphere is contractible.[2] Note this includes as a .
  • The configuration space of points in the plane is a , where is the pure braid group on strands.

Some further elementary examples can be constructed from these by using the fact that the product is .

A can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group G, and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy. The corresponding chain complex is given by the Dold–Kan correspondence.

Remark on constructing higher Eilenberg-Maclane spaces[edit]

There are multiple techniques for constructing higher Eilenberg-Maclane spaces. One of which is to construct a Moore space for an abelian group and iteratively kill the higher homotopy groups of since the lower homotopy groups are all trivial. This follows from the Hurewicz theorem.

Another useful technique is to first construct for every group using simplicial techniques,[3] and then construct the higher Eilenberg-Maclane spaces using homotopy cofibers. Note that for non-abelian ,

since all higher homotopy groups are abelian. The higher groups can be constructed using the because we can recursively use the homotopy cofiber of the fibration

to construct , giving a fibration sequence

which can be used to study the cohomology of from using the Leray spectral sequence. This was exploited by Jean-Pierre Serre while he studied the homotopy groups of spheres using the Postnikov system and spectral sequences.

One other technique is to use the geometric realization of simplicial abelian groups.[4] This gives an explicit presentations of simplicial abelian groups which represent Eilenberg-Maclane spaces. Another simplicial construction, in terms of classifying spaces and universal bundles, is given in J. Peter May's book.[5]

Properties of Eilenberg–MacLane spaces[edit]

Bijection between homotopy classes of maps and cohomology[edit]

An important property of is that, for any abelian group G, and any CW-complex X, the set

of homotopy classes of maps from X to is in natural bijection with the n-th singular cohomology group

of the space X. Thus one says that the are representing spaces for cohomology with coefficients in G. Since

there is a distinguished element corresponding to the identity. The above bijection is given by pullback of that element — . This is similar to the Yoneda lemma of category theory.

Another version of this result, due to Peter J. Huber, establishes a bijection with the n-th Čech cohomology group when X is Hausdorff and paracompact and G is countable, or when X is Hausdorff, paracompact and compactly generated and G is arbitrary. A further result of Kiiti Morita establishes a bijection with the n-th numerable Čech cohomology group for an arbitrary topological space X and G an arbitrary abelian group.

Loop spaces[edit]

The loop space of an Eilenberg–MacLane space is also an Eilenberg–MacLane space: . This property implies that Eilenberg–MacLane spaces with various n form an omega-spectrum, called an Eilenberg–MacLane spectrum. This spectrum corresponds to the standard homology and cohomology theory.

Functoriality[edit]

It follows from the universal coefficient theorem for cohomology that the Eilenberg MacLane space is a quasi-functor of the group; that is, for each positive integer if is any homomorphism of abelian groups, then there is a non-empty set

satisfying where denotes the homotopy class of a continuous map and

Relation with Postnikov tower[edit]

Every CW-complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration whose fibers are Eilenberg–MacLane spaces.

Cohomology operations[edit]

The cohomology groups of Eilenberg–MacLane spaces can be used to classify all cohomology operations.

Applications[edit]

The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence

with the string group, and the spin group. The relevance of lies in the fact that there are the homotopy equivalences

for the classifying space , and the fact . Notice that because the complex spin group is a group extension

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space is an example of a higher group. It can be thought of the topological realization of the groupoid whose object is a single point and whose morphisms are the group . Because of these homotopical properties, the construction generalizes: any given space can be used to start a short exact sequence that kills the homotopy group in a topological group.

See also[edit]

Notes[edit]

  1. ^ Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. MR13312) In this context it is therefore conventional to write the name without a space.
  1. ^ (Papakyriakopoulos 1957)
  2. ^ "general topology - Unit sphere in $\mathbb{R}^\infty$ is contractible?". Mathematics Stack Exchange. Retrieved 2020-09-01.
  3. ^ Yin, Xi. "On Eilenberg-Maclane spaces" (PDF). Archived (PDF) from the original on 21 Aug 2018.
  4. ^ "gt.geometric topology - Explicit constructions of K(G,2)?". MathOverflow. Retrieved 2020-10-28.
  5. ^ May, J. Peter. A Concise Course in Algebraic Topology (PDF). Chapter 16, section 5: University of Chicago Press.CS1 maint: location (link)

References[edit]

Foundational articles[edit]

Cartan seminar and applications[edit]

The Cartan seminar contains many fundamental results about Eilenberg-Maclane spaces including their homology and cohomology, and applications for calculating the homotopy groups of spheres.

Computing integral cohomology rings[edit]

Applications[edit]

Other encyclopedic references[edit]