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A universal space for plus-constructions. (English) Zbl 0933.55016
This paper explores a relatively new universal construction in topology and relates it to D. Quillen’s more familiar plus construction [Actes Congr. Internat. Math. 1970, 2, 47-51 (1971; Zbl 0225.18011)]. The new one, developed by Bousfield, Dror Farjoun and others, is built up using function spaces. Given a map \(f: W\to U\), one says that a space \(L\) is \(f\)-local if the induced map of function spaces, \(\text{map}(U,L)\to \text{map}(W,L)\), is a weak homotopy equivalence. It is possible to assign to each space \(X\) an \(f\)-local space \(L_fX\) and a homotopy universal map \(X\to L_fX\). When \(f\) has the form \(W\to*\), \(L_f\) is called the \(W\)-nullification functor and is denoted by \(P_W\). Alternatively, given a fixed \(W\), the \(W\)-nullification of a space \(X\) is a map of \(X\) to a space \(P_WX\), where the based function space, \(\text{map}_*(W, P_WX)\), is weakly contractible and \(P_WX\) is initial in the homotopy category of spaces under \(X\) with this property.
Quillen’s plus construction \(X\to X^+\) with respect to the largest perfect subgroup of \(\pi_1(X)\) is a nullification functor: for any appropriate representation \(W\) of the wedge of all acyclic spaces, \(P_WX\) and \(X^+\) agree up to homotopy. Better, the authors produce such a two-dimensional Eilenberg-MacLane space \(W= K({\mathcal F},1)\), where \({\mathcal F}\) is a locally free perfect group. Moreover, using the nullity class of a group \(G\) (namely, the class of all groups \(G'\) such that \(\operatorname{Hom}(G,R)\) is trivial if and only if \(\operatorname{Hom}(G',R)\) is trivial), they show that there is precisely one nullity class of locally free perfect groups \({\mathcal F}\) such that \(K({\mathcal F},1)\)-nullification is naturally isomorphic to the plus construction.
Other results include the following. For an acyclic space \(W\), \(W\)-nullification of \(X\) is the plus construction for some perfect normal subgroup of \(\pi_1(X)\). Moreover, this feature measures the acyclicity, in that, when connected, \(W\) is acyclic if and only if \(W\)-nullification invariably is an integral homology isomorphism.

55P99 Homotopy theory
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