Is it true that every invariant on manifolds comes from cohomology groups?

Question

Question: Is it true that every invariant on manifolds comes from Homological groups?

For example, characteristics classes, Betti numbers, fundamental groups (By Hurewicz's theorem), Signature of 4d-manifolds (defined by middle cohomology group), Intersection forms (defined by middle cohomology group), etc. I think all of these are related somehow to (co)homology groups. If so why geometers trying to find new invariants?

Answer 1

No, it's not true at all. For example, there is an infinite number of diffeomorphism classes of manifolds with the same homology groups as $S^3$.

Besides, your examples are wrong.

• Characteristic classes are elements of cohomology groups. Moreover, you cannot deduce what the characteristic classes are just by knowing the homology groups. They are extra information.
• You cannot know the fundamental group of a manifold if you just know its homology groups. Yes, if $M$ is path-connected, then Hurewicz's theorem says that $H_1(M)$ is the abelianization of $M$. But 1. if $M$ is not path-connected then extracting information about the different $\pi_1(M)$ from $H_1(M)$ is difficult, because you don't know what comes from which component; 2. knowing the abelianization of a group is not enough to know the group, as e.g. there are nontrivial groups with trivial abelianization (called perfect groups).

Other invariants that are not captured by the homology groups alone include the cup product on cohomology, the signature for $4d$-manifolds, the higher homotopy groups and their Whitehead product, etc. In all cases you can cook up examples with the same homology groups but where the invariants mentioned differ.