Parker Glynn-Adey

Singularities I.1

Posted in Math by pgadey on 2013/12/04

These are some notes that I’m writing up on Gromov’s papers on singularities of maps. This first post will look at some of the introductory material in: Singularities, Expanders and Topology of Maps. Pt 1. These notes will be a partial introduction to what is going on.

The general theme of the paper is as follows: Let {f : M^m \rightarrow N^n} be a smooth map and let {\hat{\Sigma}(F) = \{ m \in M : \mathop{\operatorfont rank}\nolimits(Df)|_m < n\}} to be the singularity of {f}. Let {\Sigma = \Sigma(F) = F(\hat{\Sigma}(F))} be its image. The central question is:

Can one lower bound the topological complexity of {\Sigma} in terms of {M}? In a similar spirit, can one lower bound the topological complexity of {f^{-1}(\nu)}?

It’s important to consider the relationship between {\hat{\Sigma}} and {\Sigma} for a moment before we proceed. There is a sense in which the map {F} will complicated {\hat{\Sigma}} when it maps {M} to {N}. Let’s briefly consider an example. Let {S_{g,b}} denote the closed compact orientable genus {g} surface with {b} boundary components. {S_{0,0}} is a sphere and {S_{1,0}} is a torus.

Theorem 1 (Guth) Let {g \geq 1}. If {f : S_{g,1} \rightarrow {\mathbb R}^2} is an immersion and {f(\partial S_{g,1})} intersects itself transversally then {f(\partial S_{g,1})} has at least {2g + 2} self intersections. Moreover, this is sharp.

As a corollary consider an immersion of some large even genus {S_{2g,0}} in to the plane. There is a seperating curve {\gamma} such that {S_{2g,0} \setminus \gamma = S_{g,1} \sqcup S_{g,1}}. Generically this curve will be sent to a curve with more than {2g + 2} self intersections. Upstairs, {\gamma} is very reasonable. It’s just a simple curve. Downstairs it remains a curve but now it necessarily has a bunch of self-intersections. Now consider varying {g}. The seperating curves will have constant topology, they’re just {S^1}, but as {g} increases their topological complexity in a generic image will grow with {g}.

We can produce a map {F : S_{2g,0} \rightarrow {\mathbb R}^2} to the plane which is folded along {\gamma} and hence has {\hat{\Sigma} = \gamma} and is an immersion away from {\gamma}. Then we have the topology of {\Sigma(F)} is lower bounded in terms of the topology (genus) of {\Sigma_{2g,0}}.

We will introduce the following auxillary notions of complexity on {\Sigma}.

Definition 2 We say that a ray {\gamma : [0, \infty) \rightarrow {\mathbb R}^2} goes from {\gamma(0)} to infinity if {\gamma} leaves each compact subset of {{\mathbb R}^2} in finite time. The depth {p} in {{\mathbb R}^2 \setminus \Sigma} is the infimal number of transverse intersections of {\Sigma} and rays going from {p} to infinity. The depth of {\Sigma}, written as {\mathop{\operatorfont dep}\nolimits(\Sigma)}, is the supremal depth of a point in {{\mathbb R}^2 \setminus \Sigma}.

Definition 3 We say that {p \in \Sigma} is a {n}-fold self-crossing of {\Sigma} if {|F^{-1}(p)| = n}. We write {N_{max}} for the cardinality of the set of maximally self-crossing points. We write {N_k} for the cardinality of the set of {k}-fold self-crossing points.

The main theorem of Singularities I says that:

Theorem 4 There are 3-manifolds of arbitrarily large volume such that:

  • {|\kappa| \leq 1} where {\kappa} is sectional curvature.
  • {Inj \geq 1} where {Inj} is the injectivity radius.

and for generic smooth maps {M \rightarrow {\mathbb R}^2} one has:

  • {\mathop{\operatorfont dep}\nolimits(\Sigma) \geq C \mathop{\operatorfont vol}\nolimits(M)}
  • {N_2(\Sigma) \geq C \left[ \mathop{\operatorfont vol}\nolimits(M) \right]^2}

Note that for generic smooth maps {M \rightarrow N} one has {\dim(\hat{\Sigma}) = \dim(\Sigma) = \dim(N) - 1}, consequently we should expect to only care about {N_2} in the case above.

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