Whitehead torsion

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In mathematics, Whitehead torsion is an invariant of an h-cobordism in a Whitehead group, that is important in simple homotopy theory and surgery theory. It is named for J. H. C. Whitehead.

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[edit] Whitehead torsion

Suppose that W is an h-cobordism from M to N; this means roughly that W is a manifold with boundary the union of M and N and that W is homotopy equivalent to both M and N. The Whitehead torsion of the cobordism W is an element of the Whitehead group Wh(M) of M (see below), and is an obstruction to the cobordism being a product M×[0,1]. For W of dimension at least 6, the s-cobordism theorem says that it is the only obstruction; in fact the isomorphism classes of h-cobordisms from M to something correspond exactly to elements of the Whitehead group of M.

[edit] The Whitehead group of a group

The Whitehead group of a manifold M is equal to the Whitehead group Wh(π1(M)) of the fundamental group π1(M) of M.

If G is a group, the Whitehead group of G is defined to be the abelian group given as the quotient of GL∞(Z(G)) by the subgroup generated by elementary matrices, elements of G, and −1. It is a quotient of the K-group K_1(\mathbf{Z}[G]); this latter is called the Whitehead group of the ring \mathbf{Z}[G].

Here Z(G) is the group ring of G, and for any ring A, the group GL∞(A) is defined to be the direct limit of the finite dimensional groups \operatorname{GL}_n(A) \to \operatorname{GL}_{n+1}(A); concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. (An elementary matrix here is a transvection: one such that all main diagonal elements are 1 and there is at most one non-zero element not on the diagonal.)

The subgroup generated by elementary matrices is exactly the derived subgroup, in other words the smallest normal subgroup such that the quotient by it is abelian. The quotient of

GL∞(A)

by the group generated by elementary matrices is the K-group K1(A), which is therefore the "abelianization" of GL∞(A) (and which is sometimes called the Whitehead group of the ring A).

When A is commutative there is a natural homomorphism from K1(A) to the invertible elements A* of A, induced by taking the determinant of a matrix. In this case K1(A) splits as the sum of the group SK1(A) of elements of determinant 1, and the group A*. The group SK1(A) is hard to understand in general, but is often trivial.

The Whitehead group Wh(G) can be thought of as the group K1(Z(G)) modulo the subgroup of "obvious" elements ±G.

[edit] Geometric definition

Given CW-complex A, consider the set of all pairs of CW-complexes (X,A) such that the inclusion of A into X is a homotopy equivalence. Two pairs (X1,A) and (X2,A) are said to be equivalent, if there is a simple homotopy equivalence between X1 and X2 relative to A. The set of such equivalence classes form the Whitehead group Wh(A) of CW-complex A. The group operation is defined by taking union of X1 and X2 with common subspace A. It turns out that Wh(A)=Wh(π1A). The proof of this fact is similar to the proof of s-cobordism theorem.

[edit] Examples of Whitehead groups

  • The Whitehead group of the trivial group is trivial. Since the group ring of the trivial group is Z, we have to show that any matrix can be written as a product of elementary matrices times a diagonal matrix; this follows easily from the fact that Z is a Euclidean domain.
  • The Whitehead group of a free abelian group is trivial. This is quite hard to prove, but is important as it is used in the proof that an s-cobordism of dimension at least 6 whose ends are tori is a product.
  • The Whitehead group of a braid group (or any subgroup of a braid group) is trivial. This was proved by Farrell and Roushon.
  • The Whitehead group of the cyclic groups of orders 2, 3, 4, and 6 are trivial.
  • The Whitehead group of the cyclic group of order 5 is Z. An example of a non-trivial unit in the group ring is (1−t−t4)(1−t2−t3)=1, where t is a generator of the cyclic group of order 5. This example is closely related to the existence of units of infinite order in the ring of integers of the cyclotomic field generated by fifth roots of unity.
  • If G is a finite abelian group then K1(Z(G)) is isomorphic to the units of the group ring Z(G) under the determinant map, so Wh(G) is just the group of units of Z(G) modulo the group of "trivial units" generated by elements of G and −1.
  • It is a well-known conjecture that the Whitehead group of any torsion-free group should vanish.

[edit] Whitehead torsion of a PL manifold

[edit] See also

[edit] References

  • Cohen, A course in simple homotopy theory
  • Milnor, J. Whitehead torsion Bull. Amer. Math. Soc. 72 1966 358--426.

[edit] External links