# Whitehead torsion IV (Ex)

The aim of this exercise is to prove the following:

**Proposition 0.1.**
Let be an h-coborism between closed, connected -manifolds with with finite abelian fundamental groups of odd order. If then and are s-cobordant.

You may also wish to investigate possible extensions of this proposition.

## [edit] Comments

The following results from [Milnor1966] will be helpful. Recall that the canonical involution on the group ring of a finitely generated group induces a conjuation on the Whitehead group.

**Lemma 2.1** [Milnor1966, Lemma 6.7] **.**
If is finite abelian, then every element of is self-conjugate.

**Theorem 2.2** [Milnor1966, Duality Theorem] **.**
For any orientable h-cobordism of dimension we have

where denotes the conjugate of .

Now if and denote the inclusions, compute the Whitehead torsion of the homotopy equivalence .

Finally, you may use the following theorem of Bak.

**Theorem 2.3** c.f.[Bak1975, Theorem 1]**.**
Let be a finite group of odd order, then .

## [edit] References

- [Bak1975] A. Bak,
*Odd dimension surgery groups of odd torsion groups vanish*, Topology**14**(1975), no.4, 367–374. MR0400263 (53 #4098) Zbl 0322.57021 - [Milnor1966] J. Milnor,
*Whitehead torsion*, Bull. Amer. Math. Soc.**72**(1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104