Notes on quantum weighted projective spaces and multidimensional teardrops

TL;DR

This paper explores the structure of quantum weighted projective spaces and lens spaces, establishing principal bundle properties and computing their K-theory groups, thus advancing understanding of their algebraic and topological features.

Contribution

It demonstrates that quantum lens spaces form principal circle bundles over quantum weighted projective spaces and computes their K-theory groups, providing new insights into their algebraic topology.

Findings

Quantum lens spaces are principal circle bundles over quantum weighted projective spaces.

The $K$-groups of certain quantum weighted projective and real projective spaces are explicitly computed.

The $K_1$-group of quantum lens spaces is determined.

Abstract

It is shown that the coordinate algebra of the quantum $2n+1$-dimensional lens space $\mathcal{O}(L^{2n+1}_q(\prod_{i=0}^n m_i; m_0,\ldots, m_n))$ is a principal $\mathbb{Z}$-comodule algebra or the coordinate algebra of a circle principal bundle over the weighted quantum projective space $\mathbb{WP}^n_q(m_0,\ldots, m_n)$. Furthermore, the weighted $U(1)$-action or the $\mathbb{CZ}$-coaction on the quantum odd dimensional sphere algebra $\mathcal{O}(S^{2n+1}_q)$ that defines $\mathbb{WP}^n_q(1,m_1,\ldots, m_n)$ is free or principal. Analogous results are proven for quantum real weighted projective spaces $\mathbb{RP}^{2n}_q(m_0,\ldots, m_n)$. The $K$-groups of $\mathbb{WP}^n_q(1,\ldots, 1, m)$ and $\mathbb{RP}^{2n}_q(1,\ldots, 1,m)$ and the $K_1$-group of $L^{2n+1}_q(N; m_0,\ldots, m_n)$ are computed