Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ?

I think you run out of dimension in which you can draw when $n > 4$. Further, I do not know whether you can represent the homology in such a structural way as in the $n = 3$ or $n = 4$ case.

Needless to say is, that this is only possible when you can obtain an $M^n$ as a handlebody decomposition.


The direct analogue of Kirby diagrams exist of course but they tend to be things people have difficulty visualizing. For example, attaching a 3-handle to a 5-ball amounts to embedding an $S^2$ in $S^4$. But embeddings of $S^2$ in $S^4$ are things that don't have easy diagramatic representations. There is Kamada's "Braids and knot theory in dimension 4", Carter and Saito's "Knotted surfaces" book and a few other techniques. An example that gets at the heart of the problem is that there are known knots in $S^4$ called Cappell-Shaneson knots but as of yet, nobody has found an explicit way of visualizing them in $S^4$.

In high dimensions people tend to prefer describing their manifolds via their invariants, rather than in terms of explicit constructions. This is because of the s-cobordism theorem, which in a sense tells you that that's all you really need.

Conversely, the usage of handle / surgery descriptions in dimension 4 is largely due to the lack of an s-cobordism theorem in that range.


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