In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3\_{\Bbb Z}\to \Bbb Z$, where $\Theta^3\_{\Bbb Z}$ is the group of homology cobordism classes of integral homology 3-spheres with connected sum as operation. (<https://arxiv.org/pdf/math/9903083.pdf>)


For Seifert homology spheres $\Sigma(a\_1,\dots,a\_n)$, there are some examples whose $h$-invariants are known. But is there a theorem or algorithm or formula concerning about a way to compute the $h$-invariant for a general Seifert homology sphere $\Sigma(a\_1,\dots,a\_n)$?


For the Ozsvath-Szabo $d$-invariant, a group homomorphism $\Theta^3\_{\Bbb Z}\to 2\Bbb Z$ (which is also defined using Floer theory), there are some ways to calculate it for Seifert homology spheres (e.g. <https://arxiv.org/pdf/math/0310083.pdf>), but I cannot find any results for computing the $h$-invariant for Seifert homology spheres.



Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?
No, not yet. The state of the art in computing instanton homology for Seifert spaces is in [this paper](https://arxiv.org/abs/2010.03800). This is like knowing how to compute $\widehat{HF}(\Sigma)$, which is not enough: you also want to understand $\widehat{HF}\_{red}(\Sigma)$. More or less equivalently, you want comparable results for equivariant versions along the lines of the computation of $HF^+(\Sigma)$ for all Seifert spaces.


Such a computation does not exist in the literature nor do I believe it is forthcoming in the immediate future. You might get in touch with one of the authors of that paper. I know John Baldwin has put some thought into the equivariant versions of instanton homology and may be able to say whether or not the necessary calculations exist in his head.


