In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a...

algebraic topology, there is a distinguished class of spectra called Eilenberg–MacLane spectra H A {\displaystyle HA} for any abelian group A {\displaystyle...

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and...

CW complexes. Cohomology of CW complexes is representable by an Eilenberg–MacLane space, so by the Yoneda lemma a cohomology operation of type ( n , q...

Samuel Eilenberg. Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville. He was christened "Leslie Saunders MacLane", but...

Such a space is called an Eilenberg–MacLane space. This space has the remarkable property that it is a classifying space for cohomology: there is a natural...

X are trivial; that is, BG is an Eilenberg–MacLane space, specifically a K(G, 1). An example of a classifying space for the infinite cyclic group G is...

_{n}(\mathbf {R} ^{2})} are Eilenberg–MacLane spaces of type K ( π , 1 ) {\displaystyle K(\pi ,1)} , that the unordered configuration space of the plane UConf...

Eilenberg (1848–1927), German composer Eilenberg–MacLane space Eilenberg–Moore algebra Eilenberg–Steenrod axioms Eilenberg machine Eilenburg Eulenberg (disambiguation)...

A and natural number i, there is a CW complex K(A,i) called an Eilenberg–MacLane space and a cohomology class u in Hi(K(A,i),A) such that the resulting...

group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} . For n different from 2, a...

mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of homotopy...

singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of...

analytic space Drinfeld's symmetric space Eilenberg–Mac Lane space Euclidean space Fiber space Finsler space First-countable space Fréchet space Function...

alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map h {\displaystyle h} takes a homotopy class of maps...

{\displaystyle H^{n}(X,G)}  of the space X. One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with...

if and only if B is aspherical.) Each aspherical space X is, by definition, an Eilenberg–MacLane space of type K ( G , 1 ) {\displaystyle K(G,1)} , where...

{\displaystyle \phi _{n}:X_{n}\to X_{n-1}} that are fibrations with fibers Eilenberg-MacLane spaces K ( π n ( X ) , n ) {\displaystyle K(\pi _{n}(X),n)} . In short...

representing space for F is the Eilenberg–MacLane space K(A, i). This gives a means of showing the existence of Eilenberg-MacLane spaces. Since the homotopy...

k − 1 {\displaystyle S^{k}=\Sigma S^{k-1}} . Bott periodicity Eilenberg–MacLane space Free loop Fundamental group Gray's conjecture List of topologies...

with Eilenberg–MacLane spaces can be applied to any Lie group G, giving the string group String(G). The relevance of the Eilenberg-Maclane space K ( Z...

particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. An ordinary category has objects and...

rationally, i.e. after localisation of a space, H P ∞ {\displaystyle \mathbb {HP} ^{\infty }} is an Eilenberg–Maclane space K ( Q , 4 ) {\displaystyle K(\mathbb...

_{1}(S^{1})} hence C P ∞ {\displaystyle \mathbf {CP} ^{\infty }} is an Eilenberg–MacLane space, a K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} . Because of this...

}} , which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space K(Z2, 1). For each nonnegative integer q, the modulo...

^{n})=\operatorname {GL} (n,\mathbf {Z} ).} Since the torus is an Eilenberg–MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified...

sum Smash product Adjunction space Cohomotopy Cohomotopy group Brown's representability theorem Eilenberg–MacLane space Fibre bundle Möbius strip Line...

originally given by Galewski and Stern is not triangulable. Eilenberg–MacLane space Moore space (algebraic topology) "Is the universe a dodecahedron?", article...

topological space. The Hopf–Whitney theorem produces a weaker result because the fact that the higher homotopy groups of an Eilenberg–MacLane space also vanish...

group, or more precisely in the homology of the corresponding Eilenberg–MacLane space. Here the fundamental class is taken in homology with integer coefficients...