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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 Fr?yshov $h$-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.

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  • $\begingroup$ 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 $f$? $\endgroup$ – cgodfrey Apr 12 at 6:28
  • $\begingroup$ I edited the post. $\endgroup$ – O?uz ?avk Apr 12 at 11:06
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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.

If you are willing to consider spin rational homology cobordism, then of course there's the Rochlin invariant.

Finally, the phrasing of the question suggests that you are interested in smooth cobordisms. I don't know any obstructions to topological rational homology cobordism apart from some simple things about linking forms.

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  • $\begingroup$ You're right Professor Ruberman. I forgot the emphisize $spin^c$-notion because I generally compute rational homology cobordism invariants for integral homology spheres, sorry. I know gauge theoretical invariants provides obstructions to the existence of rational homology cobordisms, as you say. But is there any gauge theoretical $spin$ or $spin^c$ rational homology cobordism invariants? $\endgroup$ – O?uz ?avk Apr 13 at 10:14

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