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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 unit 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
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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 -theory" , Princeton Univ. Press (1971)

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 .

How to Cite This Entry:
Whitehead group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_group&oldid=36233