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====References====
 
====References====
 
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<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">[a1]</TD> <TD valign="top"> A.J. Berrick, "An approach to algebraic K-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">[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">[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 K-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">[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">[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 K-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">[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>
+
<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, "K-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 K-theory" ''Topology'' , '''11''' (1972) pp. 349–370</TD></TR>
 
</table>
 
</table>

Revision as of 16:44, 17 March 2023


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 K-theory" , Pitman (1982) MR649409
[a2] A.J. Berrick, "Characterization of plus-constructive fibrations" Adv. in Math. , 48 (1983) pp. 172–176 Zbl 0531.55012
[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 K-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 K-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, "K-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 K-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=52850
This article was adapted from an original article by A.J. Berrick (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article