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].
|[a1]||A.J. Berrick, "An approach to algebraic -theory" , Pitman (1982) MR649409|
|[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) MR1392221 Zbl 0842.55001|
|[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 MR1030707|
|[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 MR0574096|
|[a6]||J.-C. Hausmann, D. Husemoller, "Acyclic maps" L'Enseign. Math. , 25 (1979) pp. 53–75 MR0543552 Zbl 0412.55008|
|[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 MR0520494 Zbl 0409.57036|
|[a8]||N. Higson, "Algebraic -theory of stable -algebras" Adv. in Math. , 67 (1988) pp. 1–140 MR922140|
|[a9]||D.M. Kan, W.P. Thurston, "Every connected space has the homology of a " Topology , 15 (1976) pp. 253–258 MR0413089 Zbl 0355.55004|
|[a10]||M. Kervaire, "Smooth homology spheres and their fundamental groups" Trans. Amer. Math. Soc. , 144 (1969) pp. 67–72 MR0253347 Zbl 0187.20401|
|[a11]||R. Levi, "On finite groups and homotopy theory" , Memoirs , 118 , Amer. Math. Soc. (1995) MR1308466 Zbl 0861.55002|
|[a12]||J.-L. Loday, "-théorie algébrique et représentations de groupes" Ann. Sci. École Norm. Sup. , 9 (1976) pp. 309–377 MR0447373 Zbl 0362.18014|
|[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 MR0548805 Zbl 0401.57034|
|[a15]||D. Quillen, "Cohomology of groups" , Actes Congrès Internat. Math. , 2 , Gauthier-Villars (1973) pp. 47–51 MR0488054 MR0488055 Zbl 0249.18022 Zbl 0245.18010 Zbl 0225.55015 Zbl 0225.18011|
|[a16]||J.B. Wagoner, "Developping classifying spaces in algebraic -theory" Topology , 11 (1972) pp. 349–370|
Plus-construction. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Plus-construction&oldid=24116