Diffeomorphisms and homotopy equivalences sliced over BO(n)

Question

There are some classical results stating sufficient conditions on a manifold $\Sigma$ such that every homotopy equivalence $\Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$ is homotopic to a diffeomorphism $\Pi(\Sigma \stackrel{\simeq}{\longrightarrow} \Sigma)$ (here "$\Pi$" denotes the functor from (simplicial) smooth manifolds to homotopy types, i.e. to infinity-groupoids).

What is known about this when the homotopy equivalences are required to respect the tangent bundle structure, up to homotopy?

More specifically, I am wondering about the following:

Since $\Sigma$ is a manifold (of dimension $n$, say), so $\Pi(\Sigma)$ is canonically equipped with a map $\Pi(\tau_\Sigma) : \Pi(\Sigma) \to B GL(n) \simeq B O(n)$. One may hence consider the automorphism group $\mathrm{Aut}_{/B O(n)}(\Pi(\Sigma))$ whose elements are homotopy auto-equivalences of $\Pi(\Sigma)$ that are required to fit into a diagram

$$ \array{ \Pi(\Sigma) && \stackrel{\simeq}{\longrightarrow} && \Pi(\Sigma) \\ & \searrow &\Leftarrow_{\simeq}& \swarrow \\ && B O(n) } \,. $$

This is naturally a topological group, whose homotopy type reflects homotopies between such maps over $B O(n)$.

How close is this to the homotopy type of the diffeomorphism group? I.e. how close is

$$ B \mathrm{Diff}(\Sigma) $$

to

$$ B \mathrm{Aut}_{/B O(n)}(\Pi(\Sigma)) $$

?

That's what I am wondering. For readers tolerant of more stacky language I would like to add the following thought, which may or may not have some bearing on this:

Write $\mathbf{B}GL(n)$ for the smooth stack which is the homotopy quotient of the point by the action of the smooth group $GL(n)$. Then the smooth tangent bundle of $\Sigma$ is equivalently encoded in a morphism of smooth stacks

$$ \tau_\Sigma : \Sigma \longrightarrow \mathbf{B}GL(n) \,. $$

Since $\Pi(\mathbf{B}GL(n)) \simeq B GL(n) \simeq B O(n)$ this is such that $\Pi(\tau_\Sigma)$ is the map on bare homotopy types considered above.

Now stack homomorphisms in the higher slice topos over $\mathbf{B}GL(n)$ are equivalently local diffeomorphisms

$$ \mathrm{LocalDiffeos}(\Sigma) \simeq \mathrm{SmoothStacks}_{/\mathbf{B}GL(n)}(\Sigma, \Sigma) \,. $$

Hence when considering the diffeomorphism group of $\Sigma$ we may equivalently think of the slice automorphism group of $\tau_\Sigma$ over $\mathbf{B}GL(n)$.

From this perspective the question is to which extent one may "take $\Pi$ into the internalized slice automorphism group" to pass from

$$ \Pi(\mathbf{Aut}_{/\mathbf{B}GL(n)}(\Sigma)) = \Pi (\mathrm{Diff}(\Sigma)) $$

to

$$ \mathrm{Aut}_{/\Pi(\mathbf{B}GL(n))}(\Pi(\Sigma)) = \mathrm{Aut}_{/B O(n)}(\Pi(\Sigma)) \,. $$

Answer 1

I wanted to say I think this is a great question, though phrasing things in terms of stacks might scare off some of the people who can best answer this question. I think in general understanding the relationship between these two spaces is a very interesting and very hard problem.

Here is an example which shows that they can definitely be different.

3-dimensional Lens spaces $L(p,q)$ have trivial tangent bundles, so the group you write $Aut_{/BO(3)}(\Pi(\Sigma))$ will just be the homotopy automorphism group of the homotopy type $\Pi(\Sigma)$. So if they are homotopy equivalent, then they are homotopy equivalent over $BO(3)$.

Now we look at the classification of lens spaces (see wikipedia for example) and we see that $L(7,1)$ and $L(7,2)$ are lens spaces which are homotopy equivalent but which are not diffeomorphic. So for these manifolds we have:

$$BAut_{/BO(3)}(\Pi(L(7,1)) \simeq BAut_{/BO(3)}(\Pi(L(7,2))$$

Now what about their diffeomorphism groups? Well we can look at arXiv:math/0411016 and we see that as spaces:

$$Diff(L(7,1)) \simeq S^1 \times S^3 \sqcup S^1 \times S^3 $$

$$Diff(L(7,2)) \simeq S^1 \times S^1 \sqcup S^1 \times S^1 $$

which implies that $BDiff(L(7,1)) \not\simeq BDiff(L(7,2))$.

So this shows that for at least one of these spaces the classifying space of the diffeomorphism group differs from the classifying space of the tangential homotopy automorphism group.