Difference between revisions of "Whitehead group"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977707.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977707.png" /></td> </tr></table> | ||
− | let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977708.png" />. A matrix differing from the | + | let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977708.png" />. A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977709.png" /> generated by all elementary matrices coincides with the commutator group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777010.png" />. The commutator quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777011.png" /> is called the Whitehead group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777013.png" /> be the element corresponding to the matrix |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777014.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777014.png" /></td> </tr></table> |
Revision as of 22:15, 10 January 2015
An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let be an associative ring with unit element and let
be the group of invertible
-matrices over
. There are natural imbeddings
![]() |
goes to
![]() |
let . A matrix differing from the identity matrix in a single non-diagonal entry is called an elementary matrix. The subgroup
generated by all elementary matrices coincides with the commutator group of
. The commutator quotient group
is called the Whitehead group of the ring
. Let
be the element corresponding to the matrix
![]() |
It has order 2. The quotient group is called the reduced Whitehead group of the ring
.
Let be a multiplicative group and let
be its group ring over
. There is a natural homomorphism
coming from the inclusion of
. The quotient group
is called the Whitehead group of the group
.
Given a homomorphism of groups , there is a natural induced homomorphism
, such that
for
. Thus
is a covariant functor from the category of groups into the category of Abelian groups. If
is an inner automorphism, then
.
The Whitehead group of the fundamental group of a space is independent of the choice of a base point and is essential for the definition of an important invariant of mappings, the Whitehead torsion.
References
[1] | J.H.C. Whitehead, "Simple homotopy types" Amer. J. Math. , 72 (1950) pp. 1–57 |
[2] | J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426 |
[3] | J.W. Milnor, "Introduction to algebraic ![]() |
Comments
If is commutative, the determinant and, hence, the special linear groups
are defined. Using these instead of the
one obtains the special Whitehead group
. One has
where
is the group of units of
.
Whitehead group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_group&oldid=36233