In this article, we study homogeneous spaces Uq(2)/ϕT\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2)/_\phi \mathbb {T}$$\end{document} and Uq(2)/ψT\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2)/_\psi \mathbb {T}$$\end{document} of the compact quantum group Uq(2),q∈C\{0}\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2),\,q\in {\mathbb {C}}\setminus \{0\}$$\end{document}. The homogeneous space Uq(2)/ϕT\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2)/_\phi \mathbb {T}$$\end{document} is shown to be the braided quantum group SUq(2)\documentclass[12pt]{minimal}
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\begin{document}$$SU_q(2)$$\end{document}. The homogeneous space Uq(2)/ψT\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2)/_\psi \mathbb {T}$$\end{document} is established as a universal C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra given by a finite set of generators and relations. Its K\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {K}}$$\end{document}-groups are computed and two families of finitely summable odd spectral triples, one is Uq(2)\documentclass[12pt]{minimal}
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\begin{document}$$U_q(2)$$\end{document}-equivariant and the other is T2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {T}^2$$\end{document}-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the K\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {K}}$$\end{document}-homology group K1(C(Uq(2)/ψT))\documentclass[12pt]{minimal}
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\begin{document}$$K^1(C(U_q(2)/_\psi \mathbb {T}))$$\end{document}.
