Eilenberg-MacLane space
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) |
Eilenberg-MacLane space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-MacLane_space&oldid=16373