# Gromov's list of 7 constructions in differential topology

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order he used:

• Algebraic geometry (affine and projective varieties, ...)
• Lie groups (homogeneous spaces, ...)
• General position arguments (Morse theory, Pontryagin-Thom construction, ...)
• Solutions to PDE (Moduli spaces in gauge theory, Floer theory, ...)
• Surgery (Cut and paste techniques, ...)
• Markov processes

I realise that I only gave 6 constructions; this was the number of separate items listed on his slides, and since he failed to discuss this part, I am left to guess that he either listed two different constructions on one line, which I interpreted to be variants of the same construction, or that failed to include one altogether.

Question How does one construct a smooth manifold from Markov processes?

I asked Gromov after the talk for explanation, but due to the rudimentary nature of my Gromovian, I was unable to understand the answer. The only word I managed to parse is "hyperbolic," though I wouldn't put too much stock in that.

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Stupid naive question: Where does covering spaces, open book decompositions, triangulations, etc. fit in? If it's Morse theory, then isn't surgery Morse theory as well? –  Daniel Moskovich Jun 21 '10 at 13:46
Hi Mohammed, perhaps the Dunfield-Thurston random 3-manifolds are examples? arxiv.org/abs/math/0502567 Take an $N$-step random walk on a Cayley graph of the mapping class group of a hyperbolic surface, so as to produce a random mapping class; glue two handlebodies to get a random 3-manifold with a Heegaard splitting. This is surgery, but the randomness highlights certain features (random 3-manifold fundamental groups have many finite-index subgroups compared to groups with random balanced presentations). –  Tim Perutz Jun 21 '10 at 15:40
According to my notes, the 7 constructions Gromov listed are: triangulations & surgery; Lie groups and locally homogeneous spaces; algebraic equations; genericity & transversality; partitions and Markov spaces; solutions of elliptic variational problems; and moduli spaces. –  Maxime Bourrigan Sep 23 '10 at 11:29
+1 for `the rudimentary nature of my Gromovian'. –  HJRW Sep 23 '10 at 14:39
The video of his talk is now available from the CMI website (updated link): claymath.org/publications/video-catalogue –  j.c. Sep 30 at 13:03

Unfortunately I missed the talk, but on the other hand Gromov have just produced a new paper called

Manifolds : Where do we come from ? What are we ? Where are we going ?

It can be found on his web page. From the title I guess there could be some intersection with the talk. In particular in section 11 called Crystals, Liposomes and Drosophila Gromov is speaking about "Markov quotients". This sounds like a way to produce "spaces" (generalisation of manifolds, I guess).

http://www.ihes.fr/~gromov/PDF/manifolds-Poincare.pdf

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After I saw this posting, I asked Gromov for a copy of his talk. This is the paper he sent me. –  Deane Yang Sep 23 '10 at 12:13
I like this note, thanks! –  Changyu Guo Sep 30 at 12:01

I suspect (but am far from certain) that Gromov may be referring to the correspondence between symbolic and hyperbolic dynamics.

The idea is basically that the 0-1 matrix corresponding to the sparsity pattern of a stochastic matrix encodes a subshift of finite type or topological Markov chain. Usually, however, one goes from the hyperbolic dynamics to the Markov description via a Markov partition or section.

I am not aware of a way to go in the other direction in general, although placing certain conditions on the Markov process would facilitate the construction of a Markov partition (which can then be made as small as one likes), for which covering sets would constitute an atlas.

Update: So I did a little digging and came across a paper by Coornaert and Papadopoulos called "Symbolic coding for the geodesic ﬂow associated to a word hyperbolic group" (Manuscripta Math. 109, 465–492 (2002), DOI 10.1007/s00229-002-0321-9, PDF available here). In it the authors discuss an idea of Gromov whereby a to each "word hyperbolic group" $\Gamma$ a space with a flow defined up to orbit equivalence is given: this flow is called the geodesic flow associated to $\Gamma$. I quote:

In the case where $\Gamma$ is the fundamental group of a compact Riemannian manifold $M$ of negative curvature, then $\Gamma$ is word hyperbolic and [the geodesic flow associated to $\Gamma$] is, up to orbit equivalence, the geodesic ﬂow on the tangent bundle of $M$.

Nowhere, however, is it indicated that the space so constructed is generically a manifold. Still, this construction is quite closely associated with the ideas mentioned earlier, as the introduction to this paper points out.

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Any chance you could elaborate on your answer and explicitly say where a manifold appears? –  Deane Yang Jun 21 '10 at 14:45

There is an interview with Gromov: http://www.ihes.fr/~gromov/PDF/rtx100300391p.pdf

Q: (...) an you describe your involvement and how your mathematical and geometric insights can be useful for problems in biology?

Gromov: I can explain how I got involved in that. Back in Russia, everybody was excited by ideas of René Thom on applying mathematics to biology. My later motivation started from a mathematical angle, from hyperbolic groups. I realized that hyperbolic Markov partitions were vaguely similar to what happens in the process of cell division. So I looked in the literature and spoke to people, and I learned that there were so-called Lindenmayer systems. (...)

And his paper on the subject: "Cell Division and Hyperbolic Geometry" http://www.ihes.fr/~gromov/PDF/16%5B71%5D.pdf

I was just reading: Visions in Mathematics: GAFA 2000 Special Volume, Part I. Gromov's article in the collection, titled: "Spaces and Questions" has a subsection: "Symbolization and Randomization" which you might find interesting, he discusses "random manifolds" at length and even touches on one of the questions in his talk: assembling combinatorial manifolds out of simplices (i.e. how many triangles).

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