- Plus construction
In
mathematics , the plus construction is a method for simplifying thefundamental group of a space without changing its homology andcohomology groups. It was introduced byDaniel Quillen . Given a perfectnormal subgroup of thefundamental group of a connectedCW complex X, attach two-cells along loops in X whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.The most common application of the plus construction is in
algebraic K-theory . If R is aunital ring, we denote by GL_n(R) the group of invertible n-by-n matrices with elements in R. GL_n(R) embeds in GL_{n+1}(R) by attaching a 1 along the diagonal and 0s elsewhere. Thedirect limit of these groups via these maps is denoted GL(R) and itsclassifying space is denoted BGL(R). The plus construction may then be applied to the perfect normal subgroup E(R) of GL(R) = pi_1(BGL(R)), generated by matrices which only differ from theidentity matrix in one off-diagonal entry. For i>0, the nthhomotopy group of the resulting space, BGL(R)^+ is the nth K-group of R, K_n(R).See also
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Semi-s-cobordism References
*citation|last=Adams|year=1978|title=Infinite loop spaces|pages=82-95|isbn=0691082065
*citation|url=http://links.jstor.org/sici?sici=0003-486X%28197111%292%3A94%3A3%3C549%3ATSOAEC%3E2.0.CO%3B2-W |first=Daniel|last= Quillen|title=The Spectrum of an Equivariant Cohomology Ring: I|journal=Annals of Mathematics , 2nd Ser.|volume= 94|issue=3 |year=1971| pages= 549-572.
*citation|url=http://links.jstor.org/sici?sici=0003-486X%28197111%292%3A94%3A3%3C573%3ATSOAEC%3E2.0.CO%3B2-T |first=Daniel|last= Quillen|title=The Spectrum of an Equivariant Cohomology Ring: II|journal=Annals of Mathematics , 2nd Ser.|volume= 94|issue=3 |year=1971| pages= 573-602.
*citation|url=http://links.jstor.org/sici?sici=0003-486X%28197211%292%3A96%3A3%3C552%3AOTCAOT%3E2.0.CO%3B2-L |first=Daniel|last= Quillen|title=On the cohomology and K-theory of the general linear groups over a finite field|journal=Annals of Mathematics , 2nd Ser.|volume= 96|issue=3 |year=1972| pages= 552-586.
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