The space of r-immersions of a union of discs in R^n - Arone, Gregory; Šarčević, Franjo (MATHEMATICA SCANDINAVICA, 2024)
For a manifold M and an integer r>1, the space of r-immersions of M in R^n is defined to be the space of immersions of M in R^n such that the preimage of every point in R^n contains fewer than r points. We consider the space of r-immersions when M is a disjoint union of k m-dimensional discs, and prove that it is equivalent to the product of the r-configuration space of k points in R^n and the k^th power of the space of injective linear maps from R^m to R^n. This result is needed in order to apply Michael Weiss’s manifold calculus to the study of r-immersions. The analogous statement for spaces of embeddings is “well-known”, but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for r-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of r-immersions.