## Singularities I.1

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 be a smooth map and let to be the *singularity* of . Let be its image. The central question is:

Can one lower bound the topological complexity of in terms of ? In a similar spirit, can one lower bound the topological complexity of ?

It’s important to consider the relationship between and for a moment before we proceed. There is a sense in which the map will complicated when it maps to . Let’s briefly consider an example. Let denote the closed compact orientable genus surface with boundary components. is a sphere and is a torus.

Theorem 1 (Guth)Let . If is an immersion and intersects itself transversally then has at least self intersections. Moreover, this is sharp.

As a corollary consider an immersion of some large even genus in to the plane. There is a seperating curve such that . Generically this curve will be sent to a curve with more than self intersections. Upstairs, 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 . The seperating curves will have constant topology, they’re just , but as increases their topological complexity in a generic image will grow with .

We can produce a map to the plane which is folded along and hence has and is an immersion away from . Then we have the topology of is lower bounded in terms of the topology (genus) of .

We will introduce the following auxillary notions of complexity on .

Definition 2We say that a ray goes from to infinity if leaves each compact subset of in finite time. Thedepth inis the infimal number of transverse intersections of and rays going from to infinity. Thedepth of, written as , is the supremal depth of a point in .

Definition 3We say that is a -fold self-crossing of if . We write for the cardinality of the set of maximally self-crossing points. We write for the cardinality of the set of -fold self-crossing points.

The main theorem of Singularities I says that:

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

- where is sectional curvature.
- where is the injectivity radius.
and for generic smooth maps one has:

Note that for generic smooth maps one has , consequently we should expect to only care about in the case above.

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