# 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].

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