Definition of "transversal intersection" for piecewise linear submanifolds

Question

I'm working with knots in the PL category.

In "Surface Knots in 4-space" of Seiichi Kamada, the author states on p. 26 that the linking number of two oriented knots $K$ and $J$ in $S^3$ is the algebraic intersection number of one of the knots, say $K$ and a Seifert surface $F$ of the other one ($J$) after assuming that $K$ and $F$ "intersect transversely in some points". The signs of the intersection points are defined via pictures that show the two different cases of orientations of $K$ and $F$. Since Kamada doesn't use tangent spaces for the definition and since he claimed to consider only PL knots, I assume "transversal intersection" doesn't mean $\forall p \in K \cap F: T_pK+T_pF = T_p S^3$ in this case.

So what is the definition of a "transversal intersection of PL submanifolds (of $S^3$)?

Answer 1

I think you may be mistaken --- transversal intersection (or "transverse" intersection, since transversal is a noun...) really does mean that the two tangent spaces span $T_p S^3$. But in the case of a PL knot, you should probably assume that all intersections are in the interior of edges, not at vertices (easy by general position, assuming finitely many vertices). For PL Seifert surfaces, the same thing: intersections only in the interior of faces. Once you have those two conditions, you just need to look at the line containing the knot-edge, and the plane containing the seifert-surface-face, and make sure that they span all of 3-space.

(or, if you like, to not be transverse, you need only show that the line lies within the plane of the seifert-surface-face; that doesn't require a definition of "span" for subspaces rather than vectors...)

Answer 2

The correct definition of transversality in PL category is on page 61 of the book by Rourke and Sanderson "Introduction to PL topology" (which is a standard sourse of the foundational material on PL manifolds). Also, take a look at the paper by Armstrong and Zeeman

Transversality for piecewise linear manifolds. Topology 6 (1967) 433–466.

The definition says that PL submanifolds $P, Q$ of dimensions $p, q$ in an $n$-dimensional PL manifold $M$ are transversal if near every intersection point in $P\cap Q$, there is a PL chart on $M$ in which $P, Q$ appear as linear subspaces of dimensions $p, q$ in $R^n$ intersecting transversally.

Note that the same definition works in the smooth category where you would use a smooth chart and $P, Q$ will be smooth submanifolds of $M$. To me, this definition (in the smooth category) is better than the standard one precisely because it generalizes directly to the PL category.

Unlike in the smooth category, there are some subtleties in the PL setting regarding manifolds with boundary, take a look at the paper by Armstrong and Zeeman.

Remark 1. Defining PL transversality using tangent spaces is a bit meaningless since you want the notion to be independent of a local PL chart, while in one chart the submanifolds might look smooth while in another chart the submanifolds will not be smooth. Another example to think about is surfaces in $R^3$ where you use the standard chart on $R^3$. Then you can work out an example of transversal intersection such that no small perturbation makes the surfaces smooth near their intersection. Just take the graph of the function $(x,y)\mapsto |x|$ as one surface and the other surface the coordinate hyperplane $y=0$.

Remark 2. Incidentally, there is also a notion of topological transversality (transversality in the category of topological manifolds) and there is a theorem that ensures existence of a transversal perturbation of a given pair of topological manifolds. However, the definition and proofs are much more difficult. See

F. Quinn, Topological transversality holds in all dimensions. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 145–148.