Can you please state the statement of Morita's theorem?

@SubhankarD. In that paper he mainly proved the statement which I mentioned in the 2 nd paragraph. I think the actual statement is a bit more complicated in terms of Johnson's homomorphism. If you want I can send you the paper. Are you convinced with the proof?

Not really! In your first paragraph itself, I think you'll get back S^3 itself ,since action of an element of torelli group gets you back the same curve in the homology of a heegaard surface ! So, you don't actually get any new homology sphere , but S^3 itself! Isn't it?

@SubhankarD. Torelli group action doesn't fix any curve.

It's a trivial action on the homology. That doesn't imply it fixed curves.

Yes, I understand.But see, if you just look at the simplest Heegaard decomposition of S^3 i.e the one with genus one heegaard surface and longitude and meridian being the attaching curves , then does acting on an element of the torelli group gives back anything new? I found a paper that said that dehn twist on separating curves and bounding pair maps generate torelli group and both of these doesn't change the attaching curves,isn't it? I've very small knowledge in Mapping class group, so let me know where I'm thinking wrong.Thanks.

@SubhankarD. Here I am not restricting genus... genus one case is the trivial... the theorem says that you can constructed homology sphere from torelli element when you allowed to vary your genus.

That is, given somne homology sphere there exists some heegard g- splitting of S^3 and some torelli elemet from that g-genus surface such that , regluing that will give you the resultant manifold.

Torelli group generated by bounding pair of maps, but what Morita proved that, for constructing homology sphere all you need is torelli element genberated by sperating curves.

A link of the paper or the results you said will be appreciated. Thanks

@subhankar I'm sorry..I don't know any link. But I've read that paper sometimes back. And as I said the paper used more technical terms like Johnson homomorphism and all.

But you can ask me questions, I can clear your doubts.

Okay; Please share the name of the paper (or its arxiv link, may be) if you find it sometimes. I'd like to understand that paper. Thanks

I may email you... Prof Dan Margalit send me that paper once. I don't have my laptop with me now. But if you believe that result, is my proof make sense to you now?

please email me the name of the paper then.I'd like to follow the whole argument actually.

But is that proof clear to you?

not really; It's fine, I've very limited knowledge in Mapping class groups, so that might have to do with my lack of understanding. That is why looking at the proof and then getting back to your argument will help me understand the whole thing, I believe.