Difference between revisions of "EilenbergMacLane space"
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Revision as of 20:51, 24 March 2012
A space, denoted by , representing the functor , where is a nonnegative number, is a group which is commutative for and is the dimensional cohomology group of a cellular space with coefficients in . It exists for any such and .
The Eilenberg–MacLane space can also be characterized by the condition: for and for , where is the th homotopy group. Thus, is uniquely defined up to a weak homotopy equivalence. An arbitrary topological space can, up to a weak homotopy equivalence, be decomposed into a twisted product of Eilenberg–MacLane spaces (see Postnikov system). The cohomology groups of coincide with those of . Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .
References
[1a]  S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces" Ann. of Math. , 46 (1945) pp. 480–509 
[1b]  S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces. II" Ann. of Math. , 51 (1950) pp. 514–533 
[2]  R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) 
[4]  E.H. Spanier, "Algebraic topology" , McGrawHill (1966) 
EilenbergMacLane space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=EilenbergMacLane_space&oldid=16373