Whitehead Torsion - Examples

Examples

• 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, a 1964 result of Bass, Heller and Swan. 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. It is also the key algebraic result used in the surgery theory classification of piecewise linear manifolds of dimension at least 5 which are homotopy equivalent to a torus; this is the essential ingredient of the 1969 Kirby–Siebenmann structure theory of topological manifolds of dimension at least 5.

• 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. This was proved in 1940 by Higman. 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.

• The Whitehead group of any finite group G is finitely generated, of rank equal to the number of irreducible real representations of G minus the number of irreducible rational representations. this was proved in 1965 by Bass.

• If G is a finite abelian group then K₁(Z) is isomorphic to the units of the group ring Z under the determinant map, so Wh(G) is just the group of units of Z 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.