On quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

The notion of a quasi-holomorphic homotopy of two immersions of a $3$-manifold into a $5$-manifold extends the notion of their regular homotopy by also allowing the homotopy to pass through instances of maps with an isolated singularity around which the path of the homotopy forms a cross-cap (complex Whitney umbrella).

We describe the local form of such homotopies and explain connections with the theory of holomorphic map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$.

Our main result is a complete a description of how the fundamental group of the complement of the image of an immersion of a $3$-manifold into a $5$-manifold (i.e. its knot group) changes under a quasi-holomorphic homotopy.

As a corollary we will see that certain quasi-holomorphic homotopies of the standard embedding of the $3$-sphere into the $5$-sphere do not change its knot group.