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Indefinite Stein fillings and $$\text {PIN}(2)$$PIN(2) -monopole Floer homology
Selecta Mathematica ( IF 1.2 ) Pub Date : 2020-02-29 , DOI: 10.1007/s00029-020-0547-y
Francesco Lin

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:

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\((Y,{\ mathfrak {s}})\)\({\ 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。
更新日期:2020-02-29
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