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.
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 .