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.

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    $\begingroup$ the answer is probably negative: mathoverflow.net/q/42629/81055 $\endgroup$ – Denis T. Dec 3 '16 at 18:23
  • $\begingroup$ Here I do not require the manifold to be smooth. In fact, I am considering constructing CW complex from $S^d$'s by gluing (such as connected sums and/or fiber sums). In other word, we start with many CW complices that describe $S^d$, and we try construct all closed orientable CW complices by gluing the d-dimensional "faces". Is this possible $\endgroup$ – Xiao-Gang Wen Dec 3 '16 at 18:48

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. 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.

  • $\begingroup$ I wonder if there is a similar surgery theory for constructing CW complex? $\endgroup$ – Xiao-Gang Wen Dec 5 '16 at 22:43
  • $\begingroup$ @Xiao-GangWen Usually CW-complexes are constructed by cell attachment. There are related things (like piecewise-linear manifolds or topological manifolds) where the possibility of constructing them by surgery has been studied very carefully in the past. $\endgroup$ – Tyler Lawson Dec 7 '16 at 2:35

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