What is known about exotic spheres up to stable diffeomorphism?

Question

In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some natural number $r$.

Question: What is known about the stable diffeomorphism classification of exotic $2k$-spheres? Are they all stably diffeomorphic to the standard sphere? If not does this happen sometimes? Are there examples which are stably diffeomorphic to the standard sphere? Or are they all distinct up to stable diffeomorphism?

Here is some background about what I know so far.

Given a manifold $M$, its stable normal bundle is classified by a map $\nu:M \to BO$. This factors through a space $B_M$ called the normal $(k-1)$-type of $M$.

$$ M \to B_M \to BO$$

The map $M \to B_M$ is an isomorphism on homotopy groups $\pi_i$ with $i < k$ and is surjective on $\pi_k$. The map $B_M \to BO$ is injective on $\pi_k$ and an isomorphism on $\pi_i$ with $i>k$. The fiber is a $(k-1)$-type.

Fixing $B \to BO$, and a manifold $M$, we can consider all maps $M \to B$ which lift the stable normal bundle $M \to BO$ and realize $B$ as the normal $(k-1)$-type of $M$. These are called normal $(k-1)$-smoothings.

We can think of this as equipping the normal bundle of $M$ with a $B$-structure. As explained here the notion of $B$-structured stable diffeomorphism also makes sense.

We have the following Theorem of Kreck.

Theorem: Let $M$ and $N$ be $2k$-dimensional closed smooth manifolds with the same normal $(k-1)$-type $B$. Then two normal $(k-1)$-smoothings $(M, \theta_M)$ and $(N, \theta_N)$ are stably diffeomorphic if and only if the bordism classes of $(M, \theta_M)$ and $(N, \theta_N)$ agree in the $B$-bordism group $\Omega_{2k}^B$ and the Euler characteristics agree.

The choice of the normal smoothing can be dealt with by considering the action of the automorphisms of $B \to BO$ on the bordism group.

If $M$ is a sphere (or exotic sphere) then the normal $(k-1)$-type is $B = BO\langle k\rangle$.

Now here it describes a way to get invariants of exotic spheres using $B$-bordism (general $B$). Any framed manifold admits a $B$-structure. If $B$ is such that $[S^n, \theta] = 0 \in \Omega^B_n$ for any framing $\theta$, then there is a well-defined homomorphism

$$\eta^B: \Theta_n \to \Omega_n^B$$

which sends an exotic sphere $\Sigma$ to $[\Sigma, \theta]$ where $\theta$ is any framing. The manifold atlas website I linked to above gives some examples. One is where $B = BO\langle \ell\rangle$ with $k+1 < \ell < 2k + 2 = n + 2$. In that case $\Omega^B_n \cong \pi_n(G/O)$, this map is very interesting and related to the famous work of Kervaire-Milnor.

The case relevant to the stable diffeomorphism classification is just outside this range, however. We can re-phrase the main question as follows.

Question: When $B = BO\langle k\rangle$, what is known about the map $\eta^B: \Theta_{2k} \to \Omega^B_{2k}?$ Specifically what is the kernel?

The same website I linked above also notes that $\eta^\textrm{Spin}$ is non-trivial in dimension $8m + 2$. So I expect that this map won't be zero in general, but I am not sure.

Answer 1

The inertia group $I_M$ of a closed oriented $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.

Wall and Kosinski proved that $I_{W_g}$ is trivial in all dimensions, where $W_g := \#^g S^n \times S^n$. This resolves your main question: two exotic spheres that are stably diffeomorphic are already diffeomorphic.

I learned this from conversations with Manuel Krannich.

As you explain, it then follows from Kreck's result that the kernel of the map $\eta^B$ agrees with $\theta_{2k}$ itself.