We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

中文翻译：

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有理cobordisms和积分同源性

我们考虑有理同调 3 球面何时与透镜空间的连通和相配的有理同调问题。我们证明了由透镜空间生成的子群中的每个有理同调共边类都由透镜空间的唯一连通和表示，其第一个同源性嵌入到同一类中的任何其他元素中。作为第一个结果，我们证明了有理同调协边群的几个自然映射具有无限阶共核。进一步的结果包括在 2 桥结的连接总和的决定因素与同一索引类中的任何其他结之间的可分条件。最后，我们使用结 Floer 同源性结合我们的主要结果来阻止 Dehn 对结进行的手术与晶状体空间的合理协调。