Response to a post of Raman’s Jeevi

 This came as a part of my response to a Telugu post by Ramana Jeevi who said that there is no dictionary which defines a human being (manishi). In my response I used a couple of quotes of Barry Mazur on Grothendieck:

Possibly the same can be said for many words. Wittgenstein said some thing like ‘ that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.’ Similar ideas come in the work of the mathematician Grothendieck: ‘mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category.’

From an obituary for him:

“He admired the HuaYen Buddhists for the attention they paid to the relationships between things rather than the things themselves, in the belief that whatever notions of identity and individuality we have emerge from those relationships.”

Thinking about Grothendieck by Barry Mazur.

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An old mathematical question

 Recently, I posted the following question

There is an old paper of mine ‘Homeomorphisms of compact 3-manifolds’ in Topology 1977 (accessible online). I was trying to extend Waldhausen theory to irreducible non-orientable 3-manifolds. So these contain two sided projective planes. The case when there are no such planes was done by W. Heil in his thesis. By decomposing further one obtains 3-manifolds with projective planes in the boundary but any two sided projective plane inside is parallel to the boundary. I was able to show that these were sufficiently large except in one case: when the boundary consists of exactly two projective planes. Now much more is known and it may be possible to handle this case.
Meanwhile, Jonathan Hillman sent me a preliminary draft in which he studies Poincaré complexes using Crisp’s thesis. One of the results he obtains is: If M is a closed irreducible manifold which has an element of finite order > 1 which has infinite centraliser then M is homotopy equivalent to projective plane cross a circle.
In the above, if N is an irreducible manifold with exactly two projective planes in the boundary, then we can glue the boundary components to obtain M as above and so the answer to the question is yes by Hillman’s result. Actually, it is a homeomrphism in Hillman’s result by an old theorem of G.R.Livesay.
Here is an argument which could have been done in 1977. Take M as above and let L be the cover corresponding to Z\cross Z_2. Take the Scott Core of L, which we may assume to have incompressible boundary. If L is not closed, the boundary has at least two projective planes. Take a map to the circle to split L along a projective plane. The result of the splitting is another compact 3- manifold K with at least four projective planes in the boundary and fundamental group Z_2. Now a famous theorem of D.B.A. Epstein from his thesis in 1960 asserts that any such K has to be projective plane times an interval. This contradiction shows that L is closed and thus equal to M itself. We are done.
If this is correct, I am still in the game but I find that I do not understand my 1977 paper. It is going to be hard work and I won’t even try.

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