Let $M$ be a (possibly simply connected) compact manifold $M$. Are there always non-zero classes in the homotopy or homology of $\mathrm{Diff}(M)$ that directly arise from the topology of $M$ itself?


As an example of the type of answers I am looking for I construct non-zero classes in the homotopy and homology of the loop space $\Omega(M)$, which come from the topology of $M$.


Let $M$ be simply connected. Then there is a smallest positive dimension $d$ where $H^d(M)$ is nonzero. Hence $\pi\_d(M)\cong H\_d(M)$ is non-zero by Hurewicz' Theorem. The long exact sequence in homotopy of the pathspace fibration shows that $\pi\_{d-1}(\Omega M)\cong \pi\_d(M)$. Applying Hurewicz' Theorem again we see that $H\_{d-1}(\Omega M)\cong \pi\_{d-1}(\Omega M)\cong \pi\_d(M)\cong H\_d(M)$. Thus the homology and homotopy have non-trivial elements that come from the topology of $M$.



Non-zero homotopy/homology in diffeomorphism groups
Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map
$$
\text{Diff}(M) \to M
$$
and so cohomology classes on $M$ pull back to ones on $\text{Diff}(M)$.


If for example,  ~~$M$ admits a nowhere zero vector field, then using the associated flow one can construct a section of the above map, so cohomology classes in $M$ inject into the cohomology of the diffeomorphism group.~~ has the structure of a Lie group, then the above map has a section and the cohomology of $M%$ will inject into the cohomology of the diffeomorphism group via the section.


This would seem to do what your asking, right?


Let me remark that
the general problem constructing non-trivial cohomology classes in the diffeomorphism group is almost 50 years old and has a lot to do with higher algebraic K-theory. The early work of Hatcher and Wagoner, Hsiang et. al., Waldhausen, Igusa, Goodwillie, Weiss and Williams are some of the names that deserve to be cited in this context.


