Whitehead torsion IV (Ex)
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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. | 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 investigate possible extensions of this proposition. |
− | == | + | == Comments == |
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]]. | ||
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Now if $i \colon M \to W$ and $i' \colon M' \to W$ denote the inclusions, compute the Whitehead torsion of the homotopy equivalence $(i')^{-1} \circ i \colon M \to M'$. | 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 | + | Finally, you may use the following theorem of Bak. |
− | {{beginthm| | + | {{beginthm|Theorem|c.f.\cite{Bak1975|Theorem 1}}} |
Let $\pi$ be a finite group of odd order, then $L_{1}^s(\Zz[\pi]) = L_3^s(\Zz[\pi]) = 0$. | Let $\pi$ be a finite group of odd order, then $L_{1}^s(\Zz[\pi]) = L_3^s(\Zz[\pi]) = 0$. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
+ | [[Category:Exercises]] | ||
+ | [[Category:Exercises with solution]] |
Latest revision as of 15:05, 1 April 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 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
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