Cobordism and Kirby calculus

As Golla pointed out that since every smooth $4$-manifold has a handle decomposition, you can draw a Kirby diagram. See the following pretty nice picture from Akbulut's lecture notes (now it is a published book titled $4$-manifolds).

enter image description here

Golla listed that lots of Brieskorn spheres which are known to be bound integral or rational homology balls, i.e., they are all integral or rational homology cobordant to $S^3$.

Following the technique of Akbulut and Larson, I also recently found new Brieskorn spheres bounding rational homology balls: $\Sigma (2,4n+3,12n+7)$ and $\Sigma(3,3n+2,12n+7)$, see the preprint.

It is interesting to note that the following Brieskorn spheres bound rational homology balls but not integral homology balls (these $4$-manifolds must contain $3$-handle(s)):

  • $\Sigma(2,3,7)$, $\Sigma(2,3,19)$,

  • $\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$ for odd $n$,

  • $\Sigma(2,4n+3,12n+7)$ and $\Sigma(3,3n+2,12n+7)$ for even $n$.

On the other hand, we know that every closed oriented $3$-manifold is cobordant to $S^3$ due to the celebrated theorem of Lickorish and Wallace.

In the following picture, you can see the explicit cobordism from $\Sigma(2,3,13)$ to $\Sigma(2,3,7)$ which is constituted by adding the red $(-1)$-framed $2$-handle. (The knot pictures are from KnotInfo). Here, blow down the red one to get the right-hand side. (Of course, they are not integral homology cobordant.)

enter image description here

Note that the Brieskorn spheres $\Sigma(2,3,6n+1)$ are obtained by $(+1)$-surgery on the twist knots $(2n+2)_1$, see for example Saveliev's book pg. 49-50. Here, $6_1$ is the stevedore knot (the left knot in the figure) and $4_1$ is the figure-eight knot (the right one).


There are many examples of the sort, in effect. As far as I know, Akbulut and Kirby (Mazur manifolds, Michigan Math. J. 26 (1979)) proved that $\Sigma(2,5,7)$, $\Sigma(3,4,5)$, and $\Sigma(2,3,13)$ bound contractible 4-manifolds; their work was then extended by Casson and Harer (Some homology lens spaces which bound rational homology balls, Pacific Math. J. 96 (1981)). Stern, Fintushel-Stern, and Fickle have more examples.

I'm sure that the Akbulut-Kirby (and some of the Casson-Harer) examples were done by Kirby calculus.

Additionally, there are also examples of Brieskorn spheres bounding rational homology 4-balls (but not integral ones, because they have non-zero Rokhlin invariant). The first example was $\Sigma(2,3,7)$, and the rational ball was produced by Fintushel and Stern (A $\mu$-invariant one homology 3-sphere that bounds an orientable rational ball, in Four-manifold theory (Durham, NH, 1982) (1984)) by explicit handle moves. This has been extended further; the latest news I have are from a paper of Akbulut and Larson (Brieskorn spheres bounding rational balls, Proc. Amer. Math. Soc. 146 (2018)), where they provide two infinite families of examples: $\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$, as well as $\Sigma(2,3,19)$.

Akbulut's book 4-manifolds (Oxford University Press) contains a wealth of examples along these lines (not many more with Brieskorn spheres, I should think).

Finally, I am not aware of (but would be interested in seeing) explicit, non-trivial examples of (rational or integral) homology cobordisms between Brieskorn spheres.