Whitehead torsion IV (Ex)
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The aim of this exercise is to prove the following: | The aim of this exercise is to prove the following: | ||
{{beginthm|Proposition}} | {{beginthm|Proposition}} | ||
− | Let $(W; M, M')$ be an [[Wikipedia:h-cobordism|h-coborism]] between closed, connected $ | + | Let $(W; M, M')$ be an [[Wikipedia:h-cobordism|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. |
{{endthm}} | {{endthm}} | ||
You may also wish to possible extensions of this proposition. | You may also wish to possible extensions of this proposition. |
Revision as of 18:22, 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.
Discussion
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