Plus-construction

Quillen plus-construction

A mapping between spaces of the homotopy type of connected CW-complexes (cf. also CW-complex), which has (necessarily a perfect normal subgroup of ) and is an acyclic mapping. This means that satisfies the following, equivalent, conditions:

the homotopy fibre of is acyclic;

induces an isomorphism of integral homology and a trivial action of on ;

induces an isomorphism of homology with any local coefficient system of Abelian groups;

if has , then there is a mapping , unique up to homotopy, such that .

When is always chosen to be the maximum perfect subgroup of the fundamental group of the domain, and the mapping is taken to be a cofibration (in fact, it can be taken to be an inclusion in a space formed by the adjunction of 2- and 3-cells), this determines a functor . General references are [a6], [a1]. A fibre sequence induces a fibre sequence if and only if acts on by mappings freely homotopic to the identity; when the space is nilpotent, this condition reduces to acting trivially on [a2].

The construction, first used in [a10], was developed by D. Quillen [a15] in order to define the higher algebraic -theory of a ring as , where the infinite general linear group is the direct limit of the finite-dimensional groups , and the plus-construction is applied to its classifying space to obtain an infinite loop space (hence spectrum) [a16]. General references are [a12], [a1]. Reconciliation with other approaches to higher -theory is found in [a5], [a13]. Subsequently, similar procedures have been employed for -algebras [a8] and ring spaces [a4].

Every connected space can be obtained by the plus-construction on the classifying space of a discrete group [a9]. Thus, the construction has also been studied for its effect on the classifying spaces of other groups, for example in connection with knot theory [a14] and finite group theory [a11]. Relations with surgery theory can be found in [a7]. For links to localization theory in algebraic topology, see [a3].

References

[a1]	A.J. Berrick, "An approach to algebraic -theory" , Pitman (1982)
[a2]	A.J. Berrick, "Characterization of plus-constructive fibrations" Adv. in Math. , 48 (1983) pp. 172–176
[a3]	E. Dror Farjoun, "Cellular spaces, null spaces and homotopy localization" , Lecture Notes , 1622 , Springer (1996)
[a4]	Z. Fiedorowicz, R. Schwänzl, R. Steiner, R.M. Vogt, "Non-connective delooping of -theory of an ring space" Math. Z. , 203 (1990) pp. 43–57
[a5]	D.R. Grayson, "Higher algebraic -theory. II (after Daniel Quillen)" , Algebraic -theory (Proc. Conf. Northwestern Univ., Evanston, Ill., 1976) , Lecture Notes in Mathematics , 551 , Springer (1976) pp. 217–240
[a6]	J.-C. Hausmann, D. Husemoller, "Acyclic maps" L'Enseign. Math. , 25 (1979) pp. 53–75
[a7]	J.-C. Hausmann, P. Vogel, "The plus-construction and lifting maps from manifolds" , Proc. Symp. Pure Math. , 32 , Amer. Math. Soc. (1978) pp. 67–76
[a8]	N. Higson, "Algebraic -theory of stable -algebras" Adv. in Math. , 67 (1988) pp. 1–140
[a9]	D.M. Kan, W.P. Thurston, "Every connected space has the homology of a " Topology , 15 (1976) pp. 253–258
[a10]	M. Kervaire, "Smooth homology spheres and their fundamental groups" Trans. Amer. Math. Soc. , 144 (1969) pp. 67–72
[a11]	R. Levi, "On finite groups and homotopy theory" , Memoirs , 118 , Amer. Math. Soc. (1995)
[a12]	J.-L. Loday, "-théorie algébrique et représentations de groupes" Ann. Sci. École Norm. Sup. , 9 (1976) pp. 309–377
[a13]	D. McDuff, G.B. Segal, "Homotopy fibrations and the "group completion" theorem" Invent. Math. , 31 (1976) pp. 279–284
[a14]	W. Meier, "Acyclic maps and knot complements" Math. Ann. , 243 (1979) pp. 247–259
[a15]	D. Quillen, "Cohomology of groups" , Actes Congrès Internat. Math. , 2 , Gauthier-Villars (1973) pp. 47–51
[a16]	J.B. Wagoner, "Developping classifying spaces in algebraic -theory" Topology , 11 (1972) pp. 349–370