zbMATH — the first resource for mathematics

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.

MSC:
 55P99 Homotopy theory
Keywords:
nullification; acyclic
Full Text: