It is possible to specify (smooth, oriented, compact) four-manifolds with special framed links:

enter image description here

$$I \times \mathbb{R}P^3$$

Such a link is divided in two sublinks, one ordinary link representing the 2-handles (and their attaching curves) and an unlinked, unknotted link representing the 1-handles.

This notation is used e.g. in Eugénia César de Sá, A link calculus for 4-manifolds (1979), and in more modern works like the book by Gompf & Stipsicz (chapter 5.4), where it is attributed to Selman Akbulut (in works from 1977 to 1979), who in turn thanks Robion Kirby and cites his influential A calculus for framed links in $S^3$ (back then yet to appear), which doesn't talk about 1-handles in 4-manifolds though.

What intrigues me is that de Sá and Akbulut don't cite each other, so I'm somehow suspecting that maybe the notion was developed by someone else (Kirby? Rourke?) and de Sá and Akbulut used it without particularly citing it because everyone knew where it came from?.

Or is the situation yet different and Akbulut "only" developed special framed links and de Sá developed "only" the $\Gamma$-moves, which tell are the equivalence transformations of special framed links?


Here is my answer, hope it clarifies the question.

A Link Calculus for 4 Manifolds was taken from part of my PhD Thesis in Warwick, U.K, with the title Automorphisms of 3 Manifolds and Representations of 4 Manifolds, 1977, which, obviously, was written before.

This article is from the proceedings of a Conference in Low Dimensional Topology that took place in Sussex 1977.

I knew the Kirby Calculus for 3 Manifolds, both from preprints that were circulating and from Kirby because I attended several seminars given by him. I also knew the representation of a 1 handle as as circle with a dot from the same source. And that's all.

Kirby is referred in both my thesis and in this article. There was a lot of "common knowledge" for orientable manifolds at the time.

All the other is original from my thesis (but I was only an unknown PhD student...)

The name Gamma moves that appear in the framed links for non-orientable manifolds and you refer to, comes from my name. A short name for Eugénia is, in my country, Gena, so I put Gamma because it's the greek letter for G.

Eugénia César de Sá

  • $\begingroup$ Thank you very much for the clarification! Is it possible to read your thesis online somewhere? $\endgroup$ – Turion Oct 13 '17 at 13:58
  • $\begingroup$ No idea. Back in those days there was nothing online. I had to typewrite the thesis and draw all the pictures by hand ( it' in colour,) $\endgroup$ – Eugénia César de Sá Oct 13 '17 at 16:53
  • $\begingroup$ No idea. Back in those days thesis were typewritten, drawings by hand . I know that copies of the thesis circulated in Princeton, late seventies. (Someone from there told me some years ago). Perhaps Warwick University library has a digital copy now. I also would like to know it. $\endgroup$ – Eugénia César de Sá Oct 13 '17 at 16:58
  • $\begingroup$ Can try to scan the 4 Manifold part of my thesis ( have to find the originals ☺) and send you by mail, if you want to. Or photopcopy them and post them to you $\endgroup$ – Eugénia César de Sá Oct 13 '17 at 21:16
  • $\begingroup$ Kirby invented framed links for 3 manifolds. Akbulut was his Phd student , I was Rourke's student. More or less at the same time so it's quite natural we don't refer to each other, because simply we didn't know each other.I was in the U. K. he was in the U. S. A. I met Kirby for the first time in 74, in Portugal. Then went to Warwick to do my PhD, and Kirby was a common visitor. So as I told before, I learned the basic concepts from him. $\endgroup$ – Eugénia César de Sá Oct 15 '17 at 18:53

Can try to scan the 4 Manifold part of my thesis ( have to find the originals ☺) and send you by mail, if you want to. Or photopcopy them and post them to you


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.