Let us consider the complex hypersurface and suppose that $n\geq 3$: $$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$ and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ of its unique singular point: $O=(0,\ldots,0)$.

We know that $V(d,n)$ is a $(2n-1)$-dimensional, $(n-2)$-connected smooth manifold (see Milnor's book "Singular points of complex hypersurfaces").

$(2n-1)$-dimensional, $(n-2)$-connected smooth manifolds have been classified by Wall, they can be written as connected sum of simple building blocks.

$\textbf{Edit:}$ Wall's classification depends on (in non exceptional cases):

1) The quadratic structure on $H^*(M)$,

2) The tangential information classified by three classes: $\alpha\in H^n(M,\pi_{n-1}(SO))$, $\beta\in H^{n+1}(M,\pi_n(SO))$ and a third element $\phi\in H^n(M,\mathbb{Z}/2)$.

$\textbf{Question:}$ Do we know this decomposition for $V(d,n)$ and what are the references?

Let me give some examples and remarks:

. When $n=3$ we know that $V(d,3)$ is diffeomorphic to a connected sum of $S^2\times S^3$.

. $V(2,n)$ is the sphere bundle of the tangent bundle of $S^n$.

$\textbf{Edit 3:}$ From wall's classification we get the decompositions

. When n is odd $V(d,n)$ can be written as a connected sum of manifolds $$M_1 + M_2 + \ldots +M_k$$ where $H_{n-1}(M_i)=\mathbb{Z}$.

. When n is even $V(d,n)$ can be written as a connected sum of manifolds $$M_1 + M_2 + \ldots +M_k'+M_{\mathbb{Z}/d}$$ where $H_{n-1}(M_i)=\mathbb{Z}$ and $H_{n-1}(M_{\mathbb{Z}/d})=\mathbb{Z}/d$.

. Moreover Each $M_i$ and $M_{\mathbb{Z}/d}$ are stably parallelizable.