I've been learning and better articulating this question over time, so my apologies to those who've read earlier iterations.
First a little background: I've been learning about mappings between spaces and I'm particularly interested in a covering map of the type:
$$\#_{k}S^{1}\times S^{2}\rightarrow S^{3}$$
I know that this is a branched covering map because each $S^{1}\times S^{2}$ is essentially the three-sphere with a handle added to it, such that $\#_{k}S^{1}\times S^{2}$ is a three-sphere with $k$ handles added to it. In trying to understand this mapping I've come across a few enlightening theorems:
First there's a more famous theorem called Alexander's theorem :
“Every closed orientable 3-manifold is a branched covering space of $S^{3}$ with branch set a link in $S^{3}$."
What I'm interested in is whether each handle has the image of a separate link on the three-sphere or whether they map to the same link. A little more searching yields another theorem:
“Every closed orientable 3-manifold is an irregular 3-sheeted branched covering of S^{3}with branch set a knot. There are two points in the inverse image of each branch point, one of which is ordinary.”
So I can take my class of manifolds and do a covering over $S^{3}$ with branch set a Link or knot. I'm wondering about each handle separately though and whether each produces a separate knot or link (depending on the map). The same paper later on, has the following theorem:
THEOREM 10. Every closed orientable 3-manifold M of genus g is an irregular 3-fold branched covering of S3 branched over a g + 2 bridge link with g +2 components each one of which is trivial, as a knot.
Here I'm finally starting to get what I'm looking for, it seems like each handle contributes an unknot to the link.
Now in my application (physics), the three-manifold represents a spacelike surface at different moments in time, each connected by a Lorentz cobordism (see here if interested: Gibbons, Hawking ).
I'm interested in what microscopic wormholes (our handles) look like to an observer measuring them assuming simply connected space (hence the mapping to the three-sphere). Here's my question: Is anyone aware of any cases where knots/links in space have the behavior of a spinorial field? in some ways their behavior appears the same (non-trivial monodramy).
Also, a point of confusion for me when the author says the inverse image of each branch point has two points, one of which is ordinary, what does that mean exactly? I'm asking because I'm interested in whether fermionlike spinor fields can "come up" in any way from these mappings.
I'll also add that I've found a theorem regarding four-dimensional space that:
“Theorem 4.Every simply connected, closed four-manifold $M$ admits a degree two map by a product which is given by the composition of a branched double covering $$S^{1}\times\left(\#_{k}S^{2}\times S^{1}\right)\rightarrow\#_{k}\mathbb{CP}^{2}\#_{k}\bar{\mathbb{CP}^{2}}$$ followed by a collapsing map $$\#_{k}\mathbb{CP}^{2}\#_{k}\bar{\mathbb{CP}^{2}}\rightarrow M$$”
I bring this up, because the complex projective space has been used to represent spinors and also corresponds the the 3-dimensional projective Hilbert space. If anyone can explain how these images of knots and links can be represented as fermionlike spinorial fields or are similar to them I would greatly appreciate it. I would also be happy with research references here.