Whitehead torsion IV (Ex)

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The aim of this exercise is to prove the following:

Proposition 0.1. Let (W; M, M') be an h-coborism between closed, connected 2k-manifolds with with finite abelian fundamental groups of odd order. If k \geq 3 then M and M' 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 \Zz[\pi] of a finitely generated group \pi induces a conjuation on the Whitehead group.

Lemma 2.1 [Milnor1966, Lemma 6.7] . If \pi is finite abelian, then every element of \text{Wh}(\pi) is self-conjugate.

Theorem 2.2 [Milnor1966, Duality Theorem] . For any orientable h-cobordism (W; M, M') of dimension n we have

\displaystyle  \tau(W, M') = (-1)^{n-1} \hat \tau(W, M)

where \hat \tau denotes the conjugate of \tau.

Now if i \colon M \to W and i' \colon M' \to W denote the inclusions, compute the Whitehead torsion of the homotopy equivalence (i')^{-1} \circ i \colon M \to M'.

Finally, you may use the following theorem of Bak.

Theorem 2.3 c.f.[Bak1975, Theorem 1]. Let \pi be a finite group of odd order, then L_{1}^s(\Zz[\pi]) = L_3^s(\Zz[\pi]) = 0.

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