We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that $\beta=-\bar{\mu}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\alpha, \beta,$ and $\gamma$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\Sigma(a_1,...,a_n)$ are not homology cobordant to any $-\Sigma(b_1,...,b_n)$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer spectrum provides homology cobordism obstructions distinct from $\alpha,\beta,$ and $\gamma$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

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塞弗特纤维的 Pin(2)-等变 Seiberg-Witten Floer 同源性

我们根据 Heegaard Floer 同源性计算 Seifert 有理同源三球体的 $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer 同源性。作为这个计算的结果，我们证明了 Manolescu 的猜想，即对于 Seifert 积分同调三球体 $\beta=-\bar{\mu}$。我们证明了 Manolescu 不变量 $\alpha,\beta,$ 和 $\gamma$ 为 Seifert 纤维空间之间的同源坐标提供了新的障碍，并且许多 Seifert 同源球 $\Sigma(a_1,...,a_n)$ 是与任何 $-\Sigma(b_1,...,b_n)$ 不同源。然后我们使用相同的不变量来给出一个积分同调球体的例子，而不是与任何 Seifert 纤维空间同调的。我们还表明，$\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer 谱提供了不同于 $\alpha,\beta,$ 和 $\gamma$ 的同源协同障碍。