In the paper "Clifford modules" by Atiyah, Bott and Shapiro, a homomorphism $\alpha:A_k\rightarrow \tilde{KO}(S^k)$ from a certain group of Clifford modules to stable homotopy groups of spheres is constructed. The composite of this homomorphism with the stable $J$-homomorphism $J:\tilde{KO}(S^k)\rightarrow\pi_{k-1}^s$ gives a homomorphism $J\circ\alpha:\alpha:A_k\rightarrow\pi_{k-1}^s$.

My question is the following: Is there a direct homomorphism $\beta : A_k\rightarrow \rightarrow\pi_{k-1}^s$ from Clifford modules to stable homotopy without mentioning vector bundles and $J$-homomorphism in the literature?

Any hint or idea about this question and reference suggestions about these topics would be greatly appreciated.

In the paper "Clifford modules" by Atiyah, Bott and Shapiro, a homomorphism $\alpha:A_k\rightarrow \tilde{KO}(S^k)$ from a certain group of Clifford modules to real $K$-theory of spheres is constructed. The composite of this homomorphism with the stable $J$-homomorphism $J:\tilde{KO}(S^k)\rightarrow\pi_{k-1}^s$ gives a homomorphism $J\circ\alpha:\alpha:A_k\rightarrow\pi_{k-1}^s$ from $A_k$ to stable homotopy groups of spheres.

My question is the following: Is there a direct homomorphism $\beta : A_k\rightarrow \rightarrow\pi_{k-1}^s$ from Clifford modules to stable homotopy without mentioning vector bundles and $J$-homomorphism in the literature?

Any hint or idea about this question and reference suggestions about these topics would be greatly appreciated.