On the Vanishing of the Rokhlin Invariant

J. Math. Sci. Univ. Tokyo
Vol. 18 (2011), No. 2, Page 239--268.

Moriyama, Tetsuhiro
On the Vanishing of the Rokhlin Invariant
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It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant $\mu(M)$ of an amphichiral integral homology $3$--sphere $M$ vanishes. In this paper, we give a new direct proof of this vanishing property. For such an $M$, we construct a manifold pair $(Y,Q)$ of dimensions $6$ and $3$ equipped with some additional structure ($6$--dimensional spin $e$-manifold), such that $Q \cong M \amalg M \amalg (-M)$, and $(Y,Q) \cong (-Y,-Q)$. We prove that $(Y,Q)$ bounds a $7$--dimensional spin $e$--manifold $(Z,X)$ by studying the cobordism group of $6$--dimensional spin $e$-manifolds and the $\Z/2$--action on the two--point configuration space of $M \setminus \left\{ pt \right\}$. For any such $(Z,X)$, the signature of $X$ vanishes, and this implies $\mu(M) = 0$. The idea of the construction of $(Y,Q)$ comes from the definition of the Kontsevich--Kuperberg--Thurston invariant for rational homology $3$--spheres.

Mathematics Subject Classification (2010): Primary 57M27, Secondary 57N70, 57R20, 55R80.
Received: 2011-05-19