Eilenberg-MacLane space
From Encyclopedia of Mathematics
A space, denoted by 
, representing the functor 
, where 
is a non-negative 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" , McGraw-Hill (1966) |
How to Cite This Entry:
Eilenberg-MacLane space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-MacLane_space&oldid=22371
Eilenberg-MacLane space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-MacLane_space&oldid=22371
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article