Whitehead torsion IV (Ex)

The aim of this exercise is to prove the following:

Proposition 0.1. Let $(W; M, M')$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(W; M, M')$ be an h-coborism between closed, connected $2k$$2k$-manifolds with with finite abelian fundamental groups of odd order. If $k \geq 3$$k \geq 3$ then $M$$M$ and $M'$$M'$ 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 $\Zz[\pi]$$\Zz[\pi]$ of a finitely generated group $\pi$$\pi$ induces a conjuation on the Whitehead group.

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

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

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

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

Now if $i \colon M \to W$$i \colon M \to W$ and $i' \colon M' \to W$$i' \colon M' \to W$ denote the inclusions, compute the Whitehead torsion of the homotopy equivalence $(i')^{-1} \circ i \colon M \to M'$$(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$$\pi$ be a finite group of odd order, then $L_{1}^s(\Zz[\pi]) = L_3^s(\Zz[\pi]) = 0$$L_{1}^s(\Zz[\pi]) = L_3^s(\Zz[\pi]) = 0$.

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