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Stream: learning: questions

Topic: submersion stable under pullback

Markus Lohmayer (Sep 30 2023 at 19:35):

The nLab page on subersions says "While the category Diff of (finite dimensional) smooth manifolds does not have all pullbacks, the pullback along a submersion always exists. This is because a submersion is transversal to every other smooth map into its codomain. Moreover, submersions are stable under pullback." Why are submersions stable under pullback? I like the recent paper by David Weisbart and Adam Yassine, but it does not answer my question. While I would greatly appreciate an explanation, a pointer to a proof in some article/book would be even better. Thanks!

John Baez (Sep 30 2023 at 21:16):

I don't know a reference, but I would try to prove like this. A smooth map between manifolds By definition $f: X \to Y$ gives for each point $x \in X$ a linear map, the differential $df: T_x X \to T_y Y$. By definition $f: X \to Y$ is a submersion iff for each $x \in X$ the differential $df: T_x X \to T_y Y$ is an epimorphism.

A commutative square of smooth maps together with any point in the upper left corner (if you now what I mean) induces a commutative square of linear maps. I suspect that if this commutative square of smooth maps is a pullback square, it induces a pullback square of linear maps.

If the last statement is true we're done, because in the category of vector spaces and linear maps, epimorphisms are stable under pullback.

ʇɐ (Oct 01 2023 at 15:00):

I don’t know a reference off the top of my head either, but here’s a sketch proof: Since a submersion is transversal to any other map, the fiber product of it and any other map exists (see e.g. Lang’s Fundamentals of differential geometry (1999: pp. 31–32)). By the constant rank theorem, every submersion is a smooth fiber bundle (i.e. locally a product projection) (EDIT 2023-10-02: not (globally) a fiber bundle of course!), and product projections are stable under pullback, so pullbacks of submersions are locally product projections (and hence submersions).

Markus Lohmayer (Oct 01 2023 at 20:47):

Thanks @John Baez and @ʇɐ for your answers! I get the basic idea. Projections from a product are epic and @John Baez said that epimorphisms in Vect are preserved under pullback. Why is this? (So far I only learned how monos/epis are preserved under pullback/pushout)

Patrick Nicodemus (Oct 01 2023 at 20:50):

The first one is a theorem about category theory, the second one is a theorem about Vect. Do you want to try proving it and show us what you tried? do you have an idea where to start

Patrick Nicodemus (Oct 01 2023 at 20:51):

Actually the projection $0\times X\to X$ is not epic in Set, so this cannot be a general theorem of category theory. But for vector spaces this is always true.

Patrick Nicodemus (Oct 01 2023 at 20:53):

Honestly I would use the fact that epics are equivalent to surjections in Vec.

Patrick Nicodemus (Oct 01 2023 at 20:54):

and use explicit known constructions for the pullback.

David Michael Roberts (Oct 02 2023 at 00:34):

It's not true that a submersion is a fibre bundle: take the map from (-2,1) disjoint union (-1,2) to (-2,2) that is inclusion on each factor.

Markus Lohmayer (Oct 02 2023 at 06:20):

@Patrick Nicodemus Thanks to your help! I managed to convince myself by means of looking at the elements/vectors. At the start I thought if might again have something to do with an universal property and necessarily got stuck, as your counterexample nicely shows.

David Michael Roberts (Oct 02 2023 at 08:11):

It really is a statement about charts on manifolds. A submersion $p\colon X\to Y$ has the property that around every point $x\in X$, there is chart that looks like $U\times V$, where $V$ is a chart in $Y$ around $y=p(x)$ and $U$ is a chart in $p^{-1}(y)$ around $x$. Further, in this chart in $X$, the map $p$ looks like the projection $U\times V\to V$ (this is the underlying map of the linear projection $E\oplus F\to F$, where these are the vector spaces containing the open sets that are the charts. This means that on the topological space that is the pullback in $\mathbf{Top}$, one can build charts around each point.

David Michael Roberts (Oct 02 2023 at 08:19):

So given $f\colon M\to Y$, and a point $(m,x)\in M\times_Y X$, the chart around it looks like $U\times W$, where $U$ is the chart around $x$ in the fibre from before, and $W$ is a chart around $m \in M$ small enough so that $f\big|_W$ lands inside $V$. The map down to $M$ is, on this new chart, a projection (and in this way, you can show it is a submersion), and the projection map to $X$ is given on the chart by the product $\mathrm{id}_U\times f\big|_W\colon U\times W \to U\times V$. These two maps are smooth.

One has to check all the details here, of course, but this is how it goes.