We can join two $S^d$'s via a connected sum to obtain $S^d\#S^d=S^d$. We can also deform a single $S^d$ via a self-connected sum (ie add a handle) to obtain $\#\_\text{self}S^d=S^{d-1}\times S^1$. 


When $d=2$, the self-connected sums can generate all 2-dimensional closed oriented manifolds from $S^2$. Do we have a similar result for higher dimensions? ie can all d-dimensional closed orientable manifolds be constructed via connected sums and self-connected sums from many $S^d$'s?


If the answer is no, then we may replace connected sum by fiber sum: can all d-dimensional closed orientable manifolds be constructed via fiber sums and self-fiber sums from many $S^d$'s?


Although I asked the question in terms of manifold, here I do not require the manifold to be smooth. In other words,we may replace the term "manifold" in the above by CW complex. 



Constructing manifolds via generalized connected sums or fiber sums
Obtaining a manifold from $S^d$ by iterative connected sum is not always possible. The homology groups of the $d$-manifold $M$ change in a controlled way when taking such a connected sum with itself to obtain a manifold $M'$: we have
$$
H\_k(M') = \begin{cases}
H\_k(M) &\text{if }k \neq 1,d-1\\
H\_k(M) \oplus \Bbb Z &\text{if }k = 1, d-1.
\end{cases}
$$
In particular, for $d > 2$ we can never obtain manifolds with torsion in their homology (like $\Bbb{RP}^3$) or with homology in middle dimensions (like $S^2 \times S^2$) by such a procedure.


A different procedure which you may be interested in, generalizing this, is called [surgery theory](https://en.wikipedia.org/wiki/Surgery_theory). Ordinary connected sums involve starting with an embedding $S^0 \times D^d \to M$, removing the interior to leave a boundary $S^0 \times S^{d-1}$, and then gluing back in $D^1 \times S^{d-1}$ along this boundary to obtain a new manifold. The surgery operations work in the same way, but now starting with an embedding $S^k \times D^{d-k} \to M$, drilling out the interior to leave a boundary $S^k \times S^{d-k-1}$, and gluing in $D^{k+1} \times S^{d-k-1}$ along this boundary. Doing such a surgery along a "band" $S^1 \times D^1$ in a 2-manifold can be an instructive example.


Any closed, smooth, oriented manifold can be obtained by a sequence of surgeries, and this has been a useful tool in geometric topology. The story is more complicated for topological manifolds.


