Plus-construction

Quillen plus-construction

A mapping $ {q _ {N} } : X \rightarrow {X _ {N} ^ {+} } $ between spaces of the homotopy type of connected CW-complexes (cf. also CW-complex), which has $ { \mathop{\rm Ker} } \pi _ {1} ( q _ {N} ) = N $( necessarily a perfect normal subgroup of $ \pi _ {1} ( X ) $) and is an acyclic mapping. This means that $ q _ {N} $ satisfies the following, equivalent, conditions:

the homotopy fibre $ {\mathcal A} _ {N} X $ of $ q _ {N} $ is acyclic;

$ q _ {N} $ induces an isomorphism of integral homology and a trivial action of $ \pi _ {1} ( X _ {N} ^ {+} ) $ on $ H _ {*} ( {\mathcal A} _ {N} X; \mathbf Z ) $;

$ q _ {N} $ induces an isomorphism of homology with any local coefficient system of Abelian groups;

if $ f : X \rightarrow Y $ has $ N \leq { \mathop{\rm Ker} } \pi _ {1} ( f ) $, then there is a mapping $ g : {X _ {N} ^ {+} } \rightarrow Y $, unique up to homotopy, such that $ f \simeq g \circ q _ {N} $.

When $ N $ is always chosen to be the maximum perfect subgroup $ {\mathcal P} \pi _ {1} ( X ) $ 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 $ q : X \rightarrow {X ^ {+} } $. General references are [a6], [a1]. A fibre sequence $ F \rightarrow E \rightarrow B $ induces a fibre sequence $ F ^ {+} \rightarrow E ^ {+} \rightarrow B ^ {+} $ if and only if $ {\mathcal P} \pi _ {1} ( B ) $ acts on $ F ^ {+} $ by mappings freely homotopic to the identity; when the space $ F ^ {+} $ is nilpotent, this condition reduces to $ {\mathcal P} \pi _ {1} ( B ) $ acting trivially on $ H _ {*} ( F; \mathbf Z ) $[a2].

The construction, first used in [a10], was developed by D. Quillen [a15] in order to define the higher algebraic $ K $- theory of a ring $ R $ as $ K _ {i} ( R ) = \pi _ {i} ( B { \mathop{\rm GL} } ( R ^ {+} ) ) $, where the infinite general linear group $ { \mathop{\rm GL} } ( R ) $ is the direct limit of the finite-dimensional groups $ { \mathop{\rm GL} } _ {n} ( R ) $, and the plus-construction is applied to its classifying space $ B { \mathop{\rm GL} } ( R ) $ to obtain an infinite loop space (hence spectrum) [a16]. General references are [a12], [a1]. Reconciliation with other approaches to higher $ K $- theory is found in [a5], [a13]. Subsequently, similar procedures have been employed for $ C ^ {*} $- algebras [a8] and $ A _ \infty $ 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 K-theory" , Pitman (1982) MR649409
[a2]	A.J. Berrick, "Characterization of plus-constructive fibrations" Adv. in Math. , 48 (1983) pp. 172–176 Zbl 0531.55012
[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 K-theory of an $A_{\infty}$ ring space" Math. Z. , 203 (1990) pp. 43–57 MR1030707
[a5]	D.R. Grayson, "Higher algebraic K-theory. II (after Daniel Quillen)" , Algebraic K-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 K-theory of stable $C^\star$-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 $K(\pi,1)$" 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, "K-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 K-theory" Topology , 11 (1972) pp. 349–370