Cellular Approximations Using Moore Spaces
Cellular approximations using Moore spaces JoseL.Rodr´ ´ıguez and Jer´ omeˆ Scherer ∗ Abstract For a two-dimensional Moore space M with fundamental group G,weidentify the effect of the cellularization CWM and the fiber P M of the nullification on an Eilenberg–Mac Lane space K(N,1), where N is any group: both induce on the fundamental group a group theoretical analogue, which can also be described in terms of certain universal extensions. We characterize completely M-cellular and M-acyclic spaces, in the case when M = M(Z/pk, 1). 0 Introduction Let M be a pointed connected CW-complex. The nullification functor PM and the cel- lularization functor CWM have been carefully studied in the last few years (see e.g. [8], [17], [18], [14]). These are generalizations of Postnikov sections and connective covers, where the role of spheres is replaced by a connected CW-complex M and its suspensions. This list of functors also includes plus-constructions and acyclic functors associated with a homology theory, for which M is a universal acyclic space ([2], [13], [22], [24]). Recall that a connected space X is called M-cellular if CWM X ' X, or, equivalently, if it belongs to the smallest class C(M) of spaces which contains M and is closed under homotopy equiv- alences and pointed homotopy colimits. Analogously, X is called M-acyclic if PM X '∗ or, equivalently, P M X ' X. It was shown in [14] that the class of M-acyclic spaces is the smallest class C(M) of spaces which contains M and is closed under homotopy equivalences, pointed homotopy colimits, and extensions by fibrations.
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