## 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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