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''Quillen plus-construction''
 
''Quillen plus-construction''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101401.png" /> between spaces of the [[Homotopy type|homotopy type]] of connected CW-complexes (cf. also [[CW-complex|CW-complex]]), which has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101402.png" /> (necessarily a perfect [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101403.png" />) and is an acyclic mapping. This means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101404.png" /> satisfies the following, equivalent, conditions:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101405.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101406.png" /> is acyclic;
+
the homotopy fibre $  {\mathcal A} _ {N} X $
 +
of $  q _ {N} $
 +
is acyclic;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101407.png" /> induces an isomorphism of integral [[Homology|homology]] and a trivial action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p1101409.png" />;
+
$  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 ) $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014010.png" /> induces an isomorphism of homology with any local coefficient system of Abelian groups;
+
$  q _ {N} $
 +
induces an isomorphism of homology with any local coefficient system of Abelian groups;
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014011.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014012.png" />, then there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014013.png" />, unique up to homotopy, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014014.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014015.png" /> is always chosen to be the maximum perfect subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014016.png" /> 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014017.png" />. General references are [[#References|[a6]]], [[#References|[a1]]]. A fibre sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014018.png" /> induces a fibre sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014019.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014020.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014021.png" /> by mappings freely homotopic to the identity; when the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014022.png" /> is nilpotent, this condition reduces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014023.png" /> acting trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014024.png" /> [[#References|[a2]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014025.png" />-theory]] of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014026.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014027.png" />, where the infinite general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014028.png" /> is the direct limit of the finite-dimensional groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014029.png" />, and the plus-construction is applied to its classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014032.png" />-theory is found in [[#References|[a5]]], [[#References|[a13]]]. Subsequently, similar procedures have been employed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014033.png" />-algebras [[#References|[a8]]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110140/p11014035.png" /> ring spaces [[#References|[a4]]].
+
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]]].
 
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]]].

Revision as of 08:06, 6 June 2020


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=24116
This article was adapted from an original article by A.J. Berrick (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article