2023/01/09 by Gregory Arone, Arone, Gregory, Franjo Šarčević +1
Computer Science · Mathematics · #55P42 #55R80 #57R40 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary: 18F50 #Secondary: 57R42 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2301.03131
openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an integer r≥ 2, the space of r-immersions of M in \Rn is defined to be the space of immersions of M in \Rn such that at most r-1 points of M are mapped to the same point in \Rn. The space of r-immersions lies ``between" the embeddings and the immersions. We calculate the connectivity of the layers in the homological Taylor tower for the space of r-immersions in \mathbb Rn (modulo immersions), and give conditions that guarantee that the connectivity of the maps in the tower approaches infinity as one goes up the tower. We also compare the homological tower with the homotopical tower, and show that up to degree 2r-1 there is a ``Hurewicz isomorphism" between the first non-trivial homotopy groups of the layers of the two towers.