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.
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
![\displaystyle \tau(W, M') = (-1)^{n-1} \hat \tau(W, M)](/images/math/c/8/e/c8e1a097e7329dd6e941eb2359fad7c4.png)
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
.
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