Whitehead torsion IV (Ex)
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You may also wish to possible extensions of this proposition. | You may also wish to possible extensions of this proposition. | ||
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The following results from \cite{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 [[Wikipedia:Whitehead torsion|Whitehead group]]. | The following results from \cite{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 [[Wikipedia:Whitehead torsion|Whitehead group]]. | ||
Revision as of 18:24, 8 February 2012
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 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
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 Wall and Bak.
2.3. Let be a finite group of odd order, then .
References
- [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104