# Rational homology cobordism invariants

As far as I searched, I couldn't find, but I wanna ask that:

Is there any rational homology cobordism invariant different than Ozsváth-Szabó $$d$$-invariant and Frøyshov $$h$$-invariant?

Edit:

A closed oriented $$3$$-manifold $$Y$$ is called rational homology sphere if $$H_{*}(Y,\mathbb{Q})= H_{*}(S^{3},\mathbb{Q})$$. Let $$Y_1$$ and $$Y_2$$ be rational homology spheres. They are said to be homology cobordant, if there exits a smooth compact oriented $$4$$-manifold $$X$$ with boundary $$\partial X= (-Y_1) \cup Y_2$$ such that $$H_*(X,Y_i; \mathbb Q)=0$$ for $$i=0,1$$.

Then a rational homology cobordism invariant is an invariant of rational homology spheres which does not change under homology cobordism.

• Could you be a bit more precise about what you mean by "rational homology cobordism invariant"? Do you mean if $X$ is a space and $f: M \to X$ then we have an invariant $\phi(f) \in H_*(X)$ that only depends on the (co) bordism class of $f$? Apr 12 '19 at 6:28
• I edited the post. Apr 12 '19 at 11:06

There are other gauge theoretic obstructions to the existence of rational homology cobordisms. See eg Furuta (Invent. Math., 100 (1990), 339–355 Fintushel-Stern (Topology, 26 (1987), 385–393), Matic (J. Differential Geom. 28 (1988), no. 2, 277–307) and my paper (Topology 27 (1988), no. 4, 401–414.) The two latter papers show that certain invariants $$\rho_\alpha$$ of the boundary $$3$$-manifolds associated to homomorphisms $$\alpha$$ to $$U(1)$$ must vanish for a rational homology cobordism. I believe that these kinds of obstructions are independent from the $$d$$ and $$h$$-invariants.
One thing to keep in mind in discussing the $$d$$ and $$h$$-invariants is that they are not really invariants of rational homology cobordism, but rather of $$spin^c$$ rational homology cobordism. A similar point holds for obstructions arising from $$\rho_\alpha$$ invariants, because to compare invariants of boundary components of a rational homology cobordism, you need that the representation on $$\pi_1$$ of the boundary extend over the interior. This is more or less the same point as requiring that some $$spin^c$$ structure on the boundary extends over the interior.