In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated surgeries on knots." (page 383).

Is there an analogous statement/theorem for 4-manifolds? Such as obtaining other 4-manifolds by repeating surgeries of submanifolds in $S^4$? What new 4-manifolds can be obtained in such constructions?

What are the obstruction to obtain any 4-manifolds from repeated surgeries of submanifolds in $S^4$?

References are very welcome.


If an $n$-dimensional smooth manifold $X'$ is obtained from $X$ by doing some surgeries, then there is an $(n+1)$-dimensional smooth cobordism $W$ from $X$ to $X'$; vice-versa, any handle decomposition of such a cobordism induces a sequence of surgeries.

Hence, the equivalence relation of "being obtained from one another by surgeries" is equivalent to the (oriented) cobordism relation. The statement at hand (which is usually credited to Lickorish and Wallace) can be rephrased by saying that the oriented cobordism group in dimension 3 is trivial.

In dimension 4, the oriented cobordism group is isomorphic to $\mathbb{Z}$, the isomorphism being given by the signature. For example, $\mathbb{CP}^2$ is not obtained from $S^4$, since their signature are different (1 and 0 respectively).


It is a theorem of Iwase that every 4-manifold can be obtained from a connected sum of a number of $\pm CP^2$ and $S^1\times B^3$ by surgery along tori:

Iwase, Dehn surgery along a torus T2-knot II, Japan Jour. of Math vol.16, no.2 (1990), 171-196


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