Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

TL;DR

This paper proves a filtered finiteness property for the image of the unstable Hurewicz homomorphism in homology, with applications to the bordism of immersions and the finiteness of spherical classes in certain homology groups.

Contribution

It establishes a filtered version of finiteness for the image of the unstable Hurewicz homomorphism and applies this to problems in bordism of immersions and the finiteness of spherical classes.

Findings

The image of certain homotopy groups under the Hurewicz map is finite for finite-dimensional CW complexes.

Finiteness of the image persists when replacing the space with a negative sphere.

Applications include finiteness results for the image of transfer maps from classifying spaces of Lie groups.

Abstract

After recent work of Hill, Hopkins, and Ravenel on the Kervaire invariant one problem, as well as Adams' solution of the Hopf invariant one problem, an immediate consequence of Curtis conjecture is that the set of spherical classes in $H_*Q_0S^0$ is finite. Similarly, Eccles conjecture, when specialised to $X=S^n$ with $n>0$, together with Adams' Hopf invariant one theorem, implies that the set of spherical classes in $H_*QS^n$ is finite. We prove a filtered version of the above the finiteness properties. We show that if $X$ is an arbitrary $CW$-complex such that $H_*X$ is finite dimensional then the image of the composition ${_2\pi_*}\Omega^l\Sigma^{l+2}X\to{_2\pi_*}Q\Sigma^2X\to H_*Q\Sigma^2X$ is finite; the finiteness remains valid if we formally replace $X$ with $S^{-1}$. As an immediate and interesting application, we observe that for any compact Lie group $G$ with $\dim\mathfrak{g}>1$ and for any $n>0$ the image of the composition ${_2\pi_*}Q\Sigma^{\dim\mathfrak{g}}BG_+^{[n]}\to{_2\pi_*}Q\Sigma^{\dim\mathfrak{g}}BG_+\to {_2\pi_*}Q_0S^0\to H_*Q_0S^0$ is finite where $\Sigma^{\dim\mathfrak{g}}BG_+\to S^0$ is a suitably twisted transfer map. Next, we consider work of Koschorke and Sanderson which using Thom-Pontrjagin construction provides a $1$-$1$ correspondence (a group isomorphism if $m+d>0$) $\Phi^{N,\xi}_{m,d}: \mathrm{Imm}_\xi^d(\mathbb{R}^m\times N) \longrightarrow [N_+,\Omega^{m+d}\Sigma^dT(\xi)]$. We apply work of Asadi and Eccles on computing Stiefel-Whitney numbers of immersions to show that given a framed immersion $M\to\mathbb{R}^{n+k}$ and choosing $n$ very large with respect to $d$ and $k$, all self-intersection manifolds of an arbitrary element of $\mathrm{Imm}_\xi^d(\mathbb{R}^m\times N)$ are boundary.