We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg–Witten Floer homology, and its application to the triangulation conjecture.

中文翻译：

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Floer理论及其拓扑应用

我们调查了可以与三歧管相关的Floer同源性的不同版本。我们还将讨论它们的应用，特别是关于外科手术，同源异性和有边界的四歧管的问题。然后，我们描述Floer稳定的同伦类型，相关的Pin（2）-等距Seiberg-Witten Floer同源性及其在三角猜想中的应用。