We introduce techniques to study the topology of Stein fillings of a given contact three-manifold \((Y,\xi )\) which are not negative definite. For example, given a \(\hbox {spin}^c\) rational homology sphere \((Y,{\mathfrak {s}})\) with \({\mathfrak {s}}\) self-conjugate such that the reduced monopole Floer homology group \({\textit{HM}}_{\bullet }(Y,{\mathfrak {s}})\) has dimension one, we show that any Stein filling which is not negative definite has \(b_2^+=1\) or 2, and \(b_2^-\) is determined in terms of the Frøyshov invariant. The proof of this uses \(\text {Pin}(2)\)-monopole Floer homology. More generally, we prove that analogous statements hold under certain assumptions on the contact invariant of \(\xi \) and its interaction with \(\text {Pin}(2)\)-symmetry. We also discuss consequences for finiteness questions about Stein fillings.

中文翻译：

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不定Stein填充和$$ \ text {PIN}（2）$$ PIN（2）-单极Floer同源性

我们介绍了一些技术来研究给定接触三流形\（（Y，\ xi）\）的Stein填充的拓扑，这些拓扑不是负定的。例如，给定一个\（\ hbox {spin} ^ c \）有理同源性球\（（Y，{\ mathfrak {s}}）\）与\（{\ mathfrak {s}} \\}自共轭，简化的单极Floer同源群\（{\ textit {HM}} _ {\ bullet}（Y，{\ mathfrak {s}}）\）的维数为1，我们证明了不是负定的Stein填充具有\（b_2 ^ + = 1 \）或2，并且\（b_2 ^-\）是根据Frøyshov不变量确定的。证明使用\（\ text {Pin}（2）\）-单极Floer同源性。更笼统地说，我们证明在某些假设下，类似语句对\（\ xi \）的接触不变量及其与\（\ text {Pin}（2）\）-对称性的相互作用保持不变。我们还将讨论有关Stein填充的有限性问题的后果。