Non-zero homotopy/homology in diffeomorphism groups

Question

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 $M$.

Answer 1

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.

Answer 2

If ${\rm Diff}(M)$ is contractible then the question of course has a negative answer. Examples where this happens are known in dimension three but not in higher dimensions. For $M$ a closed hyperbolic 3-manifold Gabai proved that ${\rm Diff}(M)$ has contractible components, and it was known earlier that $\pi_0{\rm Diff}(M)$ is isomorphic to the finite group of isometries of $M$ by Mostow rigidity and Waldhausen's work, so one just needs to find hyperbolic manifolds with trivial isometry group. The software package SnapPy should be able to do this. I dimly recall seeing papers giving examples, and perhaps someone can add a comment with a reference.

Answer 3

The diffeomorphism groups $\text{Diff}(M)$ are sensitive to stabilization, say replacing $M$ by $M \times [0,1]$, so the direct contribution of the homotopy type of $M$ to $\text{Diff}(M)$ can be obscure. If you instead look at the concordance = pseudoisotopy spaces $$P(M) = \text{Diff}(M \times [0,1] \ \text{rel}\ M \times \{0\}),$$ then the stabilization maps $P(M) \to P(M \times [0,1])$ get highly connected as the dimension of $M$ grows (by Kiyoshi Igusa's stability theorem), hence the low-dimensional homotopy and (co-)homology of $P(M)$ agrees with that of the stable pseudoisotopy space $$\mathscr{P}(M) = \text{colim}_n P(M \times [0,1]^n).$$ The homotopy type of $M$, being the space of points in $M$, and the homotopy type of the free loop space $\mathscr{L}M = Map(S^1, M)$, being the space of closed loops in $M$, both contribute to $\mathscr{P}(M)$, basically through maps $$\mathscr{P}(*) \times M \to \mathscr{P}(M)$$ and $$\mathscr{P}(S^1) \times \mathscr{L}M \to \mathscr{P}(M).$$ See the paper

https://projecteuclid.org/download/pdf_1/euclid.jdg/1214447541

of Tom Farrell and Lowell Jones. There is a naturally defined involution on $P(M)$, and by the work of Allen Hatcher, Michael Weiss and Bruce Williams you can use it to largely recover $\text{Diff}(M)$ from $P(M)$. A more precise statement involves the block diffeomorphism group $\widetilde{\text{Diff}}(M)$, which is quite well understood by surgery theory. The survey "Automorphisms of manifolds" by Weiss and Williams might be a good source. By the stable parametrized $h$-cobordism theorem, written up by Friedhelm Waldhausen, Bjørn Jahren and myself, the spaces $\mathscr{P}(*)$ and $\mathscr{P}(S^1)$ are very close to Waldhausen's algebraic $K$-theory spaces $A(*)$ and $A(S^1)$, which agree with the algebraic $K$-theory spaces of the ring spectra $S$ and $S[\mathbb{Z}]$, respectively. I have some papers on $K(S)$, and Lars Hesselholt has more information about $K(S[\mathbb{Z}])$. I think this is one of the main reasons to be interested in the algebraic $K$-theory of ring spectra.