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On the vanishing of the Rokhlin invariant
It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant 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 an...
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Published in: | arXiv.org 2008-07 |
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Main Author: | |
Format: | Article |
Language: | English |
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Online Access: | Get full text |
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Summary: | It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant 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 = M \cup M \cup (-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-actions on the two--point configuration space of M minus one point. For any such (Z,X), the signature of X vanishes, and this implies the vanishing of the Rokhlin invariant. The idea of the construction of (Y,Q) comes from the definition of the Kontsevich-Kuperberg-Thurston invariant for rational homology 3-spheres. |
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ISSN: | 2331-8422 |