Intrinsic convergence of the homological Taylor tower for $r$-immersions in $\mathbb{R}^n$

$\def\Rn{\mathbb{R}^n}$For an integer $r \geqslant 2$, 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 $\Rn$ (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.

Keywords

functor calculus, manifold calculus, Taylor tower, embedding, $r$-immersion, homotopy of spectra, homological convergence, partial configuration space

2010 Mathematics Subject Classification

55P42, 55R80, 57R40, 57R42

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F. Šarčević was partially supported by the grant P20 01109 (JUNTA/FEDER, UE). He is also grateful to Ismar Volić for all his support.

Received 21 January 2023

Received revised 12 September 2023

Accepted 10 October 2023

Published 2 October 2024