Difference between revisions of "Whitehead group"
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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 | ||
− | + | $$ | |
+ | \mathop{\rm GL} ( 1, A) \subset \dots \subset \mathop{\rm GL} ( n , A) \subset \dots ; | ||
+ | $$ | ||
− | + | $ 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 \| | ||
− | Given a homomorphism of groups | + | 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 | + | 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 $.
Whitehead group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_group&oldid=36233