Action of $\mathrm{M}(0,2n)$ on some kernel spaces coming from $\mathrm{SU}(2)$-TQFT

TL;DR

This paper investigates the action of the mapping class group on kernel spaces derived from $ ext{SU}(2)$-TQFT, proving the Andersen-Masbaum-Ueno conjecture for specific cases and relating kernel spaces to McMullen's homology eigenspaces.

Contribution

It proves the AMU conjecture for the 4-punctured sphere and certain pseudo-Anosovs, and identifies kernel spaces with McMullen's homology eigenspaces, linking TQFT representations to classical topology.

Findings

Proved the AMU conjecture for the 4-punctured sphere for all $N \\geq 2$.

Established a correspondence between kernel spaces and McMullen's homology eigenspaces.

Connected quantum representations with classical homological structures.

Abstract

For $n$ an even number, we study representations of the mapping class group of the $n$-punctured sphere arising from $\mathrm{SU}(2)$-TQFT when all punctures are colored by the same integer $N \geq 1$. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the $4$-punctured sphere for all $N \geq 2$. In the case $n \geq 6$ of punctures, we prove it for the pseudo-Anosovs satisfying a homological condition, namely they should act with a non trivial stretching factor on certain eigenspaces of homology of a $\frac{n}{2}$-fold branched cover considered by McMullen. The main idea is to consider the kernel space which is the kernel of the natural map from the skein module to the $\mathrm{SU}(2)$-TQFT. Our main theorem identifies, as representations of mapping class groups, certain of these kernel spaces with homology eigenspaces considered by McMullen. Our results concerning the AMU conjecture are obtained by taking appropriate limits of the quantum representations in which such a kernel space may appear as a subspace of the limit representation.