In Jaco's paper "Heegaard Splittings and Splitting Homomorphisms", he defines for a map between a surface and a bouquet of circles the notion of 'transverse to a point x', which is that the pre-image of that point under the map is a collection of disjoint loops plus a condition on product neighborhoods. I was wondering:
- Is every map from a surface to a bouquet of circles homotopic to a map that's transverse to a collection of points?
- Is there a more modern terminology for this term? And is there any good reference book/article for this idea?
Here is the paper: https://www.ams.org/journals/tran/1969-144-00/S0002-9947-1969-0253340-0/S0002-9947-1969-0253340-0.pdf
This is a standard notion in differential topology which has been in use since Thom introduced it in 1954. Guillemin--Pollack's book "Differential Topology" should suffice as a reference. Every continuous map $f: M \to N$ between smooth manifolds is homotopic to a smooth one which is homotopic to one transverse to any (countable!) family of smooth submanifolds of $N$.
There is a slight hiccup because the codomain here is not a smooth manifold. However, Jaco takes the preimages of non-wedge points, and each circle is certainly a smooth manifold. The corresponding statements above (proved with little change) are: $f: M \to \vee_{i=1}^k S^1$ is homotopic to a smooth transverse map in the sense that $$f^{-1}\big(\sqcup_{i=1}^k (-\pi,\pi)\big) \to \sqcup_{i=1}^k (-\pi,\pi)$$ is smooth and transverse to any countable family of points other than the wedge point.
(That is, you only bother to make this smooth where 'smooth' makes sense: on the manifold part of the codomain.)