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''Quillen plus-construction''

A mapping  $  {q _ {N} } : X \rightarrow {X _ {N}  ^ {+} } $
between spaces of the [[Homotopy type|homotopy type]] of connected CW-complexes (cf. also [[CW-complex|CW-complex]]), which has  $  { \mathop{\rm Ker} } \pi _ {1} ( q _ {N} ) = N $(
necessarily a perfect [[Normal subgroup|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|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|fundamental group]] of the domain, and the mapping is taken to be a [[Cofibration|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|functor]]  $  q : X \rightarrow {X  ^ {+} } $. 
General references are [[#References|[a6]]], [[#References|[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 ) $[[#References|[a2]]].

The construction, first used in [[#References|[a10]]], was developed by D. Quillen [[#References|[a15]]] in order to define the higher [[Algebraic K-theory|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|loop space]] (hence spectrum) [[#References|[a16]]]. General references are [[#References|[a12]]], [[#References|[a1]]]. Reconciliation with other approaches to higher  $  K $-
theory is found in [[#References|[a5]]], [[#References|[a13]]]. Subsequently, similar procedures have been employed for  $  C  ^ {*} $-
algebras [[#References|[a8]]] and  $  A _  \infty  $
ring spaces [[#References|[a4]]].

Every connected space can be obtained by the plus-construction on the [[Classifying space|classifying space]] of a discrete group [[#References|[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 [[#References|[a14]]] and finite group theory [[#References|[a11]]]. Relations with [[Surgery|surgery]] theory can be found in [[#References|[a7]]]. For links to localization theory in [[Algebraic topology|algebraic topology]], see [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A.J. Berrick, "An approach to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014036.png" />-theory" , Pitman (1982) {{MR|649409}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.J. Berrick, "Characterization of plus-constructive fibrations" ''Adv. in Math.'' , '''48''' (1983) pp. 172–176 {{ZBL|0531.55012}}</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Dror Farjoun, "Cellular spaces, null spaces and homotopy localization" , ''Lecture Notes'' , '''1622''' , Springer (1996) {{MR|1392221}} {{ZBL|0842.55001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> Z. Fiedorowicz, R. Schwänzl, R. Steiner, R.M. Vogt, "Non-connective delooping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014037.png" />-theory of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014038.png" /> ring space" ''Math. Z.'' , '''203''' (1990) pp. 43–57 {{MR|1030707}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D.R. Grayson, "Higher algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014039.png" />-theory. II (after Daniel Quillen)" , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014040.png" />-theory (Proc. Conf. Northwestern Univ., Evanston, Ill., 1976)'' , ''Lecture Notes in Mathematics'' , '''551''' , Springer (1976) pp. 217–240 {{MR|0574096}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.-C. Hausmann, D. Husemoller, "Acyclic maps" ''L'Enseign. Math.'' , '''25''' (1979) pp. 53–75 {{MR|0543552}} {{ZBL|0412.55008}} </TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top"> 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 {{MR|0520494}} {{ZBL|0409.57036}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> N. Higson, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014041.png" />-theory of stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014042.png" />-algebras" ''Adv. in Math.'' , '''67''' (1988) pp. 1–140 {{MR|922140}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top"> D.M. Kan, W.P. Thurston, "Every connected space has the homology of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014043.png" />" ''Topology'' , '''15''' (1976) pp. 253–258 {{MR|0413089}} {{ZBL|0355.55004}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Kervaire, "Smooth homology spheres and their fundamental groups" ''Trans. Amer. Math. Soc.'' , '''144''' (1969) pp. 67–72 {{MR|0253347}} {{ZBL|0187.20401}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R. Levi, "On finite groups and homotopy theory" , ''Memoirs'' , '''118''' , Amer. Math. Soc. (1995) {{MR|1308466}} {{ZBL|0861.55002}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> J.-L. Loday, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014044.png" />-théorie algébrique et représentations de groupes" ''Ann. Sci. École Norm. Sup.'' , '''9''' (1976) pp. 309–377 {{MR|0447373}} {{ZBL|0362.18014}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> D. McDuff, G.B. Segal, "Homotopy fibrations and the "group completion" theorem" ''Invent. Math.'' , '''31''' (1976) pp. 279–284</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> W. Meier, "Acyclic maps and knot complements" ''Math. Ann.'' , '''243''' (1979) pp. 247–259 {{MR|0548805}} {{ZBL|0401.57034}} </TD></TR>
<TR><TD valign="top">[a15]</TD> <TD valign="top"> D. Quillen, "Cohomology of groups" , ''Actes Congrès Internat. Math.'' , '''2''' , Gauthier-Villars (1973) pp. 47–51 {{MR|0488054}} {{MR|0488055}} {{ZBL|0249.18022}} {{ZBL|0245.18010}} {{ZBL|0225.55015}} {{ZBL|0225.18011}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.B. Wagoner, "Developping classifying spaces in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014045.png" />-theory" ''Topology'' , '''11''' (1972) pp. 349–370</TD></TR>
</table>