- Source: Plus construction
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.
Explicitly, if
X
{\displaystyle X}
is a based connected CW complex and
P
{\displaystyle P}
is a perfect normal subgroup of
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
then a map
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
is called a +-construction relative to
P
{\displaystyle P}
if
f
{\displaystyle f}
induces an isomorphism on homology, and
P
{\displaystyle P}
is the kernel of
π
1
(
X
)
→
π
1
(
Y
)
{\displaystyle \pi _{1}(X)\to \pi _{1}(Y)}
.
The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex
X
{\displaystyle X}
, attach two-cells along loops in
X
{\displaystyle 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
{\displaystyle R}
is a unital ring, we denote by
GL
n
(
R
)
{\displaystyle \operatorname {GL} _{n}(R)}
the group of invertible
n
{\displaystyle n}
-by-
n
{\displaystyle n}
matrices with elements in
R
{\displaystyle R}
.
GL
n
(
R
)
{\displaystyle \operatorname {GL} _{n}(R)}
embeds in
GL
n
+
1
(
R
)
{\displaystyle \operatorname {GL} _{n+1}(R)}
by attaching a
1
{\displaystyle 1}
along the diagonal and
0
{\displaystyle 0}
s elsewhere. The direct limit of these groups via these maps is denoted
GL
(
R
)
{\displaystyle \operatorname {GL} (R)}
and its classifying space is denoted
B
GL
(
R
)
{\displaystyle B\operatorname {GL} (R)}
. The plus construction may then be applied to the perfect normal subgroup
E
(
R
)
{\displaystyle E(R)}
of
GL
(
R
)
=
π
1
(
B
GL
(
R
)
)
{\displaystyle \operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))}
, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For
n
>
0
{\displaystyle n>0}
, the
n
{\displaystyle n}
-th homotopy group of the resulting space,
B
GL
(
R
)
+
{\displaystyle B\operatorname {GL} (R)^{+}}
, is isomorphic to the
n
{\displaystyle n}
-th
K
{\displaystyle K}
-group of
R
{\displaystyle R}
, that is,
π
n
(
B
GL
(
R
)
+
)
≅
K
n
(
R
)
.
{\displaystyle \pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).}
See also
Semi-s-cobordism
References
Adams, J. Frank (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 82–95, ISBN 0-691-08206-5
Kervaire, Michel A. (1969), "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society, 144: 67–72, doi:10.2307/1995269, ISSN 0002-9947, JSTOR 1995269, MR 0253347
Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: I", Annals of Mathematics, Second Series, 94 (3): 549–572, doi:10.2307/1970770, JSTOR 1970770.
Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: II", Annals of Mathematics, Second Series, 94 (3): 573–602, doi:10.2307/1970771, JSTOR 1970771.
Quillen, Daniel (1972), "On the cohomology and K-theory of the general linear groups over a finite field", Annals of Mathematics, Second Series, 96 (3): 552–586, doi:10.2307/1970825, JSTOR 1970825.
External links
"Plus-construction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Kata Kunci Pencarian:
- Jepang
- Pupuk Indonesia Holding Company
- Citi Field
- Brightline
- Menara Avian
- Kapal samudra kelas Olympic
- Proyek Strategis Nasional
- Indeks Harga Saham Gabungan
- Kōji Seo
- Grup Otomotif Beijing
- Plus construction
- Construction
- Algebraic K-theory
- Construction contract
- Barratt–Priddy theorem
- List of algebraic topology topics
- Plus 15
- Seabee
- Construction site safety
- Ashgabat
