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I see how invariants (of any kind of mathematical objects) are interesting in general, since classification is interesting, but mostly in case there is a vast variety of such objects that we do not understand well (for example knots or algebras).
However, when it comes to Homology 3-Spheres, it seems to me that the interest in their invariants mostly came from the Poincare Conjecture. Still, Now that it's been proven, people keep studying and constructing more of this kind of invariants, which leads me to the question -- What is so interesting about them? Are there "enough" different constructions of Homology Spheres that would make it interesting to classify them?

Thanks.

Paulo Henrique
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Pandora
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Even with the Poincare conjecture solved, invariants of homology $3$-spheres are a good first step in the classification of all smooth $3$-manifolds - a classification which is nearing its completing. Also, even within the last 12 months there have been important applications of invariants of homology spheres to rather deep and old problems (triangulability of higher dimensional manifolds).

It's true that with geometrisation we have an essential classification of all homology $3$-spheres, but this comes with a few caveats, mostly in the form of requiring the classification of other objects (hyperbolic groups, knots etc). So, invariants are still important if we want a finer classification than that given by geometrisation.

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Dan Rust
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