Eilenberg–MacLane space

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In mathematics, specifically algebraic topology, an Eilenberg–MacLane space[note 1] is a topological space with a single nontrivial homotopy group.

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, therefore any such space is often just called .

The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.

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 .
  • 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 connected knot or graph 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. This is a consequence of the Cartan–Hadamard theorem.
  • An infinite lens space given by the quotient of by the free action for is a . This can be shown using covering space theory and the fact that the infinite dimensional 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.
  • Correspondingly the nth unordered configuration space of is a , where denotes the n-strand braid group. [3]
  • The infinite symmetric product of a n-sphere is a . More generally is a for all Moore spaces .

Some further elementary examples can be constructed from these by using the fact that the product is . For instance the n-dimensional Torus is a .

Remark on constructing Eilenberg–MacLane spaces[edit]

For and an arbitrary group the construction of is identical to that of the classifying space of the group . Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional.

There are multiple techniques for constructing higher Eilenberg-Maclane spaces. One of which is to construct a Moore space for an abelian group : Take the wedge of n-spheres, one for each generator of the group A and realise the relations between these generators by attaching (n+1)-cells via corresponding maps in of said wedge sum. Note that the lower homotopy groups are already trivial by construction. Now iteratively kill all higher homotopy groups by successively attaching cells of dimension greater than , and define as direct limit under inclusion of this iteration.

Another useful technique is to use the geometric realization of simplicial abelian groups.[4] This gives an explicit presentation 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]

Since taking the loop space lowers the homotopy groups by one slot, we have a canonical homotopy equivalence , hence there is a fibration sequence

.

Note that this is not a cofibration sequence ― the space is not the homotopy cofiber of .

This fibration sequence 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.

Properties of Eilenberg–MacLane spaces[edit]

Bijection between homotopy classes of maps and cohomology[edit]

An important property of 's is that for any abelian group G, and any based CW-complex X, the set of based homotopy classes of based 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 singular cohomology with coefficients in G. Since

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

A constructive proof of this theorem can be found here,[6] another making use of the relation between omega-spectra and generalized reduced cohomology theories can be found here [7] and the main idea is sketched later as well.

Loop spaces / Omega spectra[edit]

The loop space of an Eilenberg–MacLane space is again an Eilenberg–MacLane space: . Further there is an adjoint relation between the loop-space and the reduced suspension: , which gives the structure of an abelian group, where the operation is the concatenation of loops. This makes the bijection mentioned above a group isomorphism.

Also this property implies that Eilenberg–MacLane spaces with various n form an omega-spectrum, called an "Eilenberg–MacLane spectrum". This spectrum defines via a reduced cohomology theory on based CW-complexes and for any reduced cohomology theory on CW-complexes with for there is a natural isomorphism , where denotes reduced singular cohomology. Therefore these two cohomology theories coincide.

In a more general context, Brown representability says that every reduced cohomology theory on based CW-complexes comes from an omega-spectrum.

Relation with Homology[edit]

For a fixed abelian group there are maps on the stable homotopy groups

induced by the map . Taking the direct limit over these maps, one can verify that this defines a reduced homology theory

on CW complexes. Since vanishes for , agrees with reduced singular homology with coefficients in G on CW-complexes.

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/Whitehead tower[edit]

Every connected CW-complex possesses a Postnikov tower, that is an inverse system of spaces:

such that for every :

  1. there are commuting maps , which induce isomorphism on for ,
  2. for ,
  3. the maps are fibrations with fiber .

Dually there exists a Whitehead tower, which is a sequence of CW-complexes:

such that for every :

  1. the maps induce isomorphism on for ,
  2. is n-connected,
  3. the maps are fibrations with fiber

With help of Serre spectral sequences computations of higher homotopy groups of spheres can be made. For instance and using a Whitehead tower of can be found here,[8] more generally those of using a Postnikov systems can be found here. [9]

Cohomology operations[edit]

For fixed natural numbers m,n and abelian groups G,H exists a bijection between the set of all cohomology operations and defined by , where is a fundamental class.

As a result, cohomology operations cannot decrease the degree of the cohomology groups and degree preserving cohomology operations are corresponding to coefficient homomorphism . This follows from the Universal coefficient theorem for cohomology and the (m-1)-connectedness of .

Some interesting examples for cohomology operations are Steenrod Squares and Powers, when are finite cyclic groups. When studying those the importance of the cohomology of with coefficients in becomes apparent quickly;[10] some extensive tabeles of those groups can be found here. [11]

Group (co)homology[edit]

One can define the group (co)homology of G with coefficients in the group A as the singular (co)homology of the Eilenberg-MacLane space with coefficients in A.

Further 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, C. D. (15 January 1957). "On Dehn's lemma and the asphericity of knots". Proceedings of the National Academy of Sciences. 43 (1): 169–172. Bibcode:1957PNAS...43..169P. doi:10.1073/pnas.43.1.169. PMC 528404. PMID 16589993.
  2. ^ "general topology - Unit sphere in $\mathbb{R}^\infty$ is contractible?". Mathematics Stack Exchange. Retrieved 2020-09-01.
  3. ^ Lucas Williams "Configuration spaces for the working undergraduate",arXiv , November 5, 2019. Retrieved 2021-06-14
  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.{{cite book}}: CS1 maint: location (link)
  6. ^ Xi Yin "On Eilenberg-MacLanes Spaces", Retrieved 2021-06-14
  7. ^ Allen Hatcher "Algebraic Topology", Cambridge University Press, 2001. Retrieved 2021-06-14
  8. ^ Xi Yin "On Eilenberg-MacLanes Spaces", Retrieved 2021-06-14
  9. ^ Allen Hatcher Spectral Sequences, Retrieved 2021-04-25
  10. ^ Cary Malkievich "The Steenrod algebra", Retrieved 2021-06-14
  11. ^ Integral Cohomology of Finite Postnikov Towers

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]

Other encyclopedic references[edit]