May I ask what is the basic motivation behind studying bounded cohomology?
Is there a simple reason why bounded cohomology is more interesting / useful than usual cohomology?
Also, is bounded cohomology only interesting for infinite spaces (with infinite simplices/singular simplices)? From the definition, it seems that for finite spaces (e.g. finite simplicial complex), the definition reduces to the usual cohomology. Is there any variant that is interesting for finite spaces?
Thanks.
The canonical homomorphism from bounded cohomology to cohomology has often nontrivial kernel. For example it is infinite-dimensional (in second cohomology) for non-elementary hyperbolic groups. So invariants that are trivial in cohomology can be nontrivial in bounded cohomology.
An example: group actions on the circle can be classified via their Euler class in bounded cohomology: there is an universal Euler class $$e\in H^2_b(Homeo^+(S^1),{\mathbf Z})$$ and every (orientation-preserving) group action on the circle, i.e., every homomorphism $$\rho\colon \Gamma\to Homeo^+(S^1)$$ gives an Euler class $$\rho^*e\in H^2_b(\Gamma,{\mathbf Z}).$$ A theorem of Ghys asserts that actions of a countable group on the circle are up to topological semi-conjugacy classified by their Euler classes as elements of $H^2_b(\Gamma,{\mathbf Z})$. No such result can be true for the Euler class in ordinary cohomology. For example $H^2({\mathbf Z},{\mathbf Z})=0$, so the Euler class of ${\mathbf Z}$-actions vanishes in ordinary homology, while the Euler class in $$H^2_b({\mathbf Z},{\mathbf Z})={\mathbf R}/{\mathbf Z}$$ gives the classical rotation number.
I would like to add a comment about the root reason why bounded cohomology was introduced (see Gromov "Volume and bounded cohomology"). The idea was to create a dual object to homology groups equipped with a natural $\ell_1$-seminorm. This seminorm was used to define simplicial volume - an invariant used in Gromov's proof of the Mostow Rigidity Theorem (see Thurson "Geometry and Topology of 3-manifolds", lecture notes, Chapter 6).
So the boundedness condition allows us to define a seminorm $$\| \cdot \|_\infty$$ on bounded cohomology groups. And there is a duality result:
$$ \| \alpha \|_1 = \sup \left\{\frac{1}{\|\phi\|_\infty} \mid \phi \in H_b^k(X), \phi(\alpha) = 1 \right\} $$
The other observation is that for spaces with amenable fundamental groups, the bounded cohomology is trivial. The heuristics behind it is that, since cocycles are bounded functions, we can average them using the mean on the group. Even more heuristics is that higher homotopy groups are abelian, and hence they are amenable, so they have no input to bounded cohomology. Hence, bounded cohomology depends only on the fundamental group of our space.
I think that one of the first papers that exploits that heuristics is Ivanov "Foundations of the theory of bounded cohomology", https://link.springer.com/article/10.1007/BF01086634.
Even though this is an old question, I want to add a different answer by saying that to me personally bounded cohomology in itself is not very interesting. It tends to be infinite dimensional or zero. What is interesting though is the comparison map $$ H^*_b(X) \to H^*(X) $$ and the quotient norm it imposes on the usual cohomology groups. Since this quotient norm behaves very naturally i.e. $$ ||\phi|| \geq ||f_*(\phi)|| $$ one obtains various restrictions on possible morphisms between cohomology rings, which is actually quite powerful. In that spirit, for example the Milnor-Wood Inequalities state that any morphism $$ H^2(\mathrm{Homeo}^+(S^1))\to H^2(\Sigma_g) $$ will send the generator to a class in $[-(2g-2)[\Sigma_g]^*,(2g-2)[\Sigma_g]^*]$. More examples are a complete classification of possible mapping degrees between surfaces, and various vanishing results for pullbacks of classes to spaces with amenable fundamental groups.
