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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.

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  • $\begingroup$ What invariants are sufficient to classify a manifold in this classification? $\endgroup$ – Will Sawin Mar 3 '14 at 2:42
  • $\begingroup$ Very interesting question. I looked at the homology computations of Brieskorn, Pham, Hirzebruch - this should at least answer the question of how many summands there are (I can give the details in an answer if you want...). One thing I am confused about: how do you get the torsion summand to be cyclic. From the computations of Brieskorn and Hirzebruch, it seems to me that e.g. for $(3,3,3,3)$ the torsion should be an elementary abelian $3$-group of rank $5$ (but I may have been making a mistake there). $\endgroup$ – Matthias Wendt Sep 15 '14 at 13:33
  • $\begingroup$ Another comment: in the case of odd $n$ you can apply the results of R. De Sapio: On $(k-1)$-connected $(2k+1)$-manifolds. Math. Scand. 25 (1970) 181-189. Up to connected sum with a homotopy sphere, the summands are $S^{n-2}\times S^{n-1}$ or the tangent sphere bundle for the $(n-1)$-sphere. Figuring out which homotopy sphere can probably be done as in Brieskorn's paper, by computation of the signature. I will keep staring at the details... $\endgroup$ – Matthias Wendt Sep 15 '14 at 13:37

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