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An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977701.png" /> be an associative ring with unit element and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977702.png" /> be the group of invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977703.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977704.png" />. There are natural imbeddings
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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/w0977705.png" /></td> </tr></table>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w0977706.png" /> goes to
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An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [[#References|[1]]]. Let  $  A $
 +
be an associative ring with unit element and let  $  \mathop{\rm GL} ( n , A ) $
 +
be the group of invertible  $  ( n \times n ) $-
 +
matrices over  $  A $.
 +
There are natural imbeddings
  
<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>
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$$
 +
\mathop{\rm GL} ( 1, A)  \subset  \dots \subset    \mathop{\rm GL} ( n , A)  \subset  \dots ;
 +
$$
  
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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$  g \in  \mathop{\rm GL} ( n, A) $
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goes to
  
<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>
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$$
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\left (
  
It has order 2. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777015.png" /> is called the reduced Whitehead group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777016.png" />.
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let  $  \mathop{\rm GL} ( A) = \cup _ {i=} 1  ^  \infty  \mathop{\rm GL} ( i, A) $.
 +
A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup  $  E( A) \subset  \mathop{\rm GL} ( A) $
 +
generated by all elementary matrices coincides with the commutator group of  $  \mathop{\rm GL} ( A) $.  
 +
The commutator quotient group $  K _ {1} A = \mathop{\rm GL} ( A) / E( A) $
 +
is called the Whitehead group of the ring $  A $.  
 +
Let  $  [- 1] \in K _ {1} A $
 +
be the element corresponding to the matrix
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777017.png" /> be a multiplicative group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777018.png" /> be its group ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777019.png" />. There is a natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777020.png" /> coming from the inclusion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777021.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777022.png" /> is called the Whitehead group of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777023.png" />.
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$$
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\left \|
  
Given a homomorphism of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777024.png" />, there is a natural induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777025.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777027.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777028.png" /> is a covariant functor from the category of groups into the category of Abelian groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777029.png" /> is an inner automorphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777030.png" />.
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It has order 2. The quotient group  $  \overline{K}\; _ {1} ( A) = K _ {1} A/ \{ 0, [- 1] \} $
 +
is called the reduced Whitehead group of the ring  $  A $.
 +
 
 +
Let  $  \Pi $
 +
be a multiplicative group and let  $  \mathbf Z [ \Pi ] $
 +
be its group ring over  $  \mathbf Z $.
 +
There is a natural homomorphism  $  j:  \Pi \rightarrow \overline{K}\; _ {1} \mathbf Z [ \Pi ] $
 +
coming from the inclusion of  $  \Pi \subset  \mathop{\rm GL} ( 1, \mathbf Z [ \Pi ]) $.
 +
The quotient group  $  \mathop{\rm Wh} ( \Pi ) = \overline{K}\; _ {1} \mathbf Z [ \Pi ] / j ( \Pi ) $
 +
is called the Whitehead group of the group  $  \Pi $.
 +
 
 +
Given a homomorphism of groups $  f : \Pi _ {1} \rightarrow \Pi _ {2} $,  
 +
there is a natural induced homomorphism $  \mathop{\rm Wh} ( f  ) :   \mathop{\rm Wh} ( \Pi _ {1} ) \rightarrow  \mathop{\rm Wh} ( \Pi _ {2} ) $,  
 +
such that $  \mathop{\rm Wh} ( g \circ f  ) = \mathop{\rm Wh} ( g) \circ  \mathop{\rm Wh} ( f  ) $
 +
for $  g : \Pi _ {2} \rightarrow \Pi _ {3} $.  
 +
Thus $  \mathop{\rm Wh} $
 +
is a covariant functor from the category of groups into the category of Abelian groups. If $  f : \Pi \rightarrow \Pi $
 +
is an inner automorphism, then $  \mathop{\rm Wh} ( f  ) = \mathop{\rm id} _ { \mathop{\rm Wh}  ( \Pi ) } $.
  
 
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|Whitehead torsion]].
 
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|Whitehead torsion]].
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====Comments====
 
====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777032.png" /> is commutative, the determinant and, hence, the special linear groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777033.png" /> are defined. Using these instead of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777034.png" /> one obtains the special Whitehead group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777035.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777036.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777037.png" /> is the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777038.png" />.
+
If $  A $
 +
is commutative, the determinant and, hence, the special linear groups $  \mathop{\rm SL} ( n, A) $
 +
are defined. Using these instead of the $  \mathop{\rm GL} ( n, A) $
 +
one obtains the special Whitehead group $  SK _ {1} ( A) $.  
 +
One has $  K _ {1} ( A) = U( A) \oplus SK _ {1} ( A) $
 +
where $  U( A) $
 +
is the group of units of $  A $.

Latest revision as of 08:29, 6 June 2020


An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let $ A $ be an associative ring with unit element and let $ \mathop{\rm GL} ( n , A ) $ be the group of invertible $ ( n \times n ) $- matrices over $ A $. There are natural imbeddings

$$ \mathop{\rm GL} ( 1, A) \subset \dots \subset \mathop{\rm GL} ( n , A) \subset \dots ; $$

$ g \in \mathop{\rm GL} ( n, A) $ goes to

$$ \left ( let $ \mathop{\rm GL} ( A) = \cup _ {i=} 1 ^ \infty \mathop{\rm GL} ( i, A) $. A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup $ E( A) \subset \mathop{\rm GL} ( A) $ generated by all elementary matrices coincides with the commutator group of $ \mathop{\rm GL} ( A) $. The commutator quotient group $ K _ {1} A = \mathop{\rm GL} ( A) / E( A) $ is called the Whitehead group of the ring $ A $. Let $ [- 1] \in K _ {1} A $ be the element corresponding to the matrix $$ \left \|

It has order 2. The quotient group $ \overline{K}\; _ {1} ( A) = K _ {1} A/ \{ 0, [- 1] \} $ is called the reduced Whitehead group of the ring $ A $.

Let $ \Pi $ be a multiplicative group and let $ \mathbf Z [ \Pi ] $ be its group ring over $ \mathbf Z $. There is a natural homomorphism $ j: \Pi \rightarrow \overline{K}\; _ {1} \mathbf Z [ \Pi ] $ coming from the inclusion of $ \Pi \subset \mathop{\rm GL} ( 1, \mathbf Z [ \Pi ]) $. The quotient group $ \mathop{\rm Wh} ( \Pi ) = \overline{K}\; _ {1} \mathbf Z [ \Pi ] / j ( \Pi ) $ is called the Whitehead group of the group $ \Pi $.

Given a homomorphism of groups $ f : \Pi _ {1} \rightarrow \Pi _ {2} $, there is a natural induced homomorphism $ \mathop{\rm Wh} ( f ) : \mathop{\rm Wh} ( \Pi _ {1} ) \rightarrow \mathop{\rm Wh} ( \Pi _ {2} ) $, such that $ \mathop{\rm Wh} ( g \circ f ) = \mathop{\rm Wh} ( g) \circ \mathop{\rm Wh} ( f ) $ for $ g : \Pi _ {2} \rightarrow \Pi _ {3} $. Thus $ \mathop{\rm Wh} $ is a covariant functor from the category of groups into the category of Abelian groups. If $ f : \Pi \rightarrow \Pi $ is an inner automorphism, then $ \mathop{\rm Wh} ( f ) = \mathop{\rm id} _ { \mathop{\rm Wh} ( \Pi ) } $.

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 $ A $ is commutative, the determinant and, hence, the special linear groups $ \mathop{\rm SL} ( n, A) $ are defined. Using these instead of the $ \mathop{\rm GL} ( n, A) $ one obtains the special Whitehead group $ SK _ {1} ( A) $. One has $ K _ {1} ( A) = U( A) \oplus SK _ {1} ( A) $ where $ U( A) $ is the group of units of $ A $.

How to Cite This Entry:
Whitehead group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_group&oldid=16652