# 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 -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
How to Cite This Entry:
Plus-construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plus-construction&oldid=48193
This article was adapted from an original article by A.J. Berrick (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article