# Non-zero homotopy/homology in diffeomorphism groups

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 $M$.

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.

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.

• Excellent, this is exactly the type of answer that I was hoping for! I'll leave the question open for a bit longer, but I am already very happy! – Thomas Rot Feb 12 '18 at 17:55
• I’m probably missing something silly, but how does a flow define a section? I can see how to define a map from R to Diff(M) or a map from M to Map(R,M), but I don’t see a map from M to Diff(M). – Dylan Wilson Feb 12 '18 at 19:11
• I don't believe this. To what diffeomorphism does your section send $x \in M$? – Oscar Randal-Williams Feb 12 '18 at 19:11
• Duh...I was a trigger happy when I wrote the above. A flow doesn't suffice. However, when $M$ is a Lie group there is a section. – John Klein Feb 13 '18 at 0:37
• To extend John's comment, the map $Diff(M) \to M$ extends to maps $Diff(M) \to C_n(M)$ and $Diff(M) \to Emb(X,M)$. In the first case you restrict the diffeomorphisms to a finite subset of $M$, in the second case you restrict the diffeomorphisms to a submanifold. Often these maps are informative. – Ryan Budney Feb 13 '18 at 2:05

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.

• I don't believe there is a proof, but I suspect it's widely believed that "most" hyperbolic 3-manifolds have trivial symmetry group. Exactly what definition of "most" one uses could perhaps change the answer. But if you look at the ratio of hyperbolic 3-manifolds with symmetry to ones without, given a bound on the volume (as the volume goes to infinity) this should be zero. If it doesn't take long I'll look through SnapPy's census. . . – Ryan Budney Feb 13 '18 at 19:27
• There are certainly hyperbolic $3$-manifolds with trivial isometry group, and in fact, for each $n>1$ every finite group is the isometry group of hyperbolic $n$-manifold by a theorem of M.Belolipetsky and A.Lubotzky, see arxiv.org/pdf/math/0406607.pdf. The 3d case is due to S.Kojima. – Igor Belegradek Feb 13 '18 at 21:36
• Thank you for this very interesting class of examples. This is very surprising to me, which probably shows my ignorance of this topic. – Thomas Rot Feb 14 '18 at 14:51

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

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.