Difference between revisions of "Plus-construction"
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''Quillen plus-construction'' | ''Quillen plus-construction'' | ||
− | A mapping | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 ![]() |
[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 ![]() ![]() |
[a5] | D.R. Grayson, "Higher algebraic ![]() ![]() |
[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 ![]() ![]() |
[a9] | D.M. Kan, W.P. Thurston, "Every connected space has the homology of a ![]() |
[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, "![]() |
[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 ![]() |
Plus-construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plus-construction&oldid=48193