Eilenberg-MacLane room

In topology , a branch of mathematics , Eilenberg-MacLane spaces are an important class of topological spaces , which on the one hand serve as building blocks in homotopy theory to assemble any CW complexes by means of fibers , and on the other hand the aspheric manifolds, which is important in differential geometry include.

definition

For a group and a natural number , a connected CW-complex is an Eilenberg-MacLane space , if for its homotopy groups ${\ displaystyle G}$ ${\ displaystyle n}$ ${\ displaystyle X}$ ${\ displaystyle K (G, n)}$

{\ displaystyle \ pi _ {n} (X) = G}

and

{\ displaystyle \ pi _ {k} (X) = 0}

for all

{\ displaystyle k \ neq 0, n}

applies.

Existence and uniqueness

If and is an Abelian group or if and is anything, then there is a CW complex that is a. ${\ displaystyle n \ geq 2}$ ${\ displaystyle G}$ ${\ displaystyle n = 1}$ ${\ displaystyle G}$ ${\ displaystyle X}$ ${\ displaystyle K (G, n)}$

This CW-complex is clearly defined except for homotopy equivalence , in the homotopy theory these CW-complexes are therefore simply referred to as "the" . ${\ displaystyle K (G, n)}$

Examples

${\ displaystyle K (G, 1)}$ - Spaces are aspherical spaces in particular ; they occur in numerous applications in mathematics.

The infinite-dimensional real-projective space is a .

{\ displaystyle \ mathbb {R} P ^ {\ infty}}

{\ displaystyle K (\ mathbb {Z} / 2 \ mathbb {Z}, 1)}

The circle is a .

{\ displaystyle S ^ {1}}

{\ displaystyle K (\ mathbb {Z}, 1)}

The bouquet of circles is one for the free group . ${\ displaystyle n}$ ${\ displaystyle K (F_ {n}, 1)}$ ${\ displaystyle F_ {n}}$

The closed, orientable area of the gender is one for the area group . ${\ displaystyle S_ {g}}$ ${\ displaystyle g \ geq 1}$ ${\ displaystyle K (\ pi _ {1} S_ {g}, 1)}$ ${\ displaystyle \ pi _ {1} S_ {g}}$

A closed, orientable, prime 3-manifold is a .

{\ displaystyle M \ not = S ^ {2} \ times S ^ {1}}

{\ displaystyle K (\ pi _ {1} M, 1)}

Every manifold of non-positive intersection curvature (and more generally every metric space whose universal superposition is a CAT (0) space ) is a . This includes locally symmetric spaces of non-compact type , especially hyperbolic manifolds . ${\ displaystyle K (G, 1)}$

${\ displaystyle K (G, n)}$ - Spaces for any play an important role in numerous applications of algebraic topology . ${\ displaystyle n}$

The infinite-dimensional complex-projective space is a . ${\ displaystyle \ mathbb {C} P ^ {\ infty}}$ ${\ displaystyle K (\ mathbb {Z}, 2)}$
The product of one and one is one . ${\ displaystyle K (G_ {1}, n)}$ ${\ displaystyle K (G_ {2}, n)}$ ${\ displaystyle K (G_ {1} \ times G_ {2}, n)}$

Postnikov decomposition

Each CW complex can be dismantled as a Postnikov tower , i. H. as iterated fibers , the fibers of which are Eilenberg-MacLane spaces. Using spectral sequences , one can then try to calculate the homotopy groups of the CW complex from the known homotopy groups of the Eilenberg-MacLane spaces.

Singular cohomology

Eilenberg-MacLane areas represent the singular cohomology is: for any topological space , each and every abelian group applies ${\ displaystyle X}$ ${\ displaystyle n}$ ${\ displaystyle G}$

{\ displaystyle H ^ {n} (X, G) = \ left [X, K (G, n) \ right],}

where the square brackets denote the set of homotopy classes of continuous mappings.

Group homology and group cohomology

The group homology of a group (with coefficients ) is by definition the singular homologue of the Eilenberg-MacLane space : ${\ displaystyle G}$ ${\ displaystyle A}$ ${\ displaystyle K (G, 1)}$

{\ displaystyle H _ {*} (G; A): = H _ {*} (K (G, 1); A),}

accordingly for group cohomology :

{\ displaystyle H ^ {*} (G; A): = H ^ {*} (K (G, 1); A).}

literature

S. Eilenberg, S. MacLane: Relations between homology and homotopy groups of spaces Ann. of Math. 46 (1945) pp. 480-509
S. Eilenberg, S. MacLane: Relations between homology and homotopy groups of spaces. II Ann. of Math. 51 (1950) pp. 514-533
Chapter 8.1 in: Edwin H. Spanier, Algebraic topology. Corrected reprint. Springer-Verlag, New York-Berlin, 1981. ISBN 0-387-90646-0