# Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?

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.

• Do you want this for something in particular? There may be tricks in specific cases to get around the lack of a nice algorithm.
– mme
Commented Oct 22, 2022 at 0:43

No, not yet. The state of the art in computing instanton homology for Seifert spaces is in this paper. 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.