A classical result of simply connected surgery theory, is that if two normal maps $f:M\_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a homotopy sphere $\Sigma$ such that $M\_0$ is diffeomorphic to $M\_1\#\Sigma$. This was first proved by Novikov in 


*Homotopically equivalent smooth manifolds. I, Izv. Akad. Nauk SSSR Ser. Mat., 28:2 (1964), 365–474.* (Theorem 5.1)


but can also be found in Browder's


*Surgery on simply-connected manifolds, Springer-Verlag (1972), Ergebnisse der Mathematik und ihre Grenzgebiete, Band 65* (II.3.7 Theorem)


and many other references of surgery theory.


My question is the following: is there a way to determine the diffeomorphism type of the added homotopy sphere?



Diffeomorphism type of the added sphere in simply connected surgery
The ambiguity in the the added homotopy sphere is captured by a suitable *inertia group $I(X)$*, which in this case is the group of homotopy spheres $\Sigma$ such that $\Sigma$ bounds a parallelizable manifold, and the standard homeomorphism $\Sigma\# X\to X$ is homotopic to the diffeomorphism.


There is a related inertial group $\bar I(V)$ that consists of homotopy spheres $\Sigma$ such that $\Sigma\# X$ and $X$ are diffeomorphic. For example, if $X=S^3\times CP^2$, then $\bar I(X)=\Theta\_7$, the group of all homotopy $7$-spheres.


The group $bP\_d$ of homotopy spheres bounding parallelizable $d$-dimensional manifolds is cyclic of known order.
More precisely, the order is known except when $d=126$, the only remaining case of the Kervaire invariant problem, see [here](https://en.wikipedia.org/wiki/Exotic_sphere). Thus to understand the ambiguity in the the added homotopy sphere one needs to compute the index of $I(X)$ in $bP\_{d}$ where $\dim(X)=d-1$.


Suppose $d$ is divisible by $4$ and $d\ge 8$, and $X$ is a closed oriented manifold of dimension $d-1$. Then $bP\_{d}$ has large order given in terms of Bernoulli numbers. For such $d$ a theorem of L.Taylor says that the index of $I(X)$ in $bP\_{d}$ is $\ge 2$. This is actually sharp, e.g., the index is $2$ for $X=S^3\times CP^{2m}$. The index is $4$ if $X=S^7\times CP^2$. On the other hand, Browder showed that
$I(X)$ is trivial when $d$ is not divisible by $8$, the group $H^1(X;\mathbb Z\_2)$ is zero, and $X$ stably parallelizable; thus in this case one can ``determine the diffeomorphism type of the added homotopy sphere". 


More details and references to the above claims can be found [here](https://arxiv.org/abs/0912.4874)
on p.8. 


