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I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...

Further edit: Thanks everyone for answers, I'm almost inclined to accept Per's answer, but I'm not still satisfied :p

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the identity map $S^n\to S^n$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the identity map $S^n\to S^n$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the identity map $S^n\to S^n$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 \circ \eta_4$. Fine.

My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta_3$ and $\eta_3\circ \eta_4$ ? How about the generator of $\pi_6(S^3)$ ?

(Other questions on MO look more serious. Hopefully this question is not out of place ...)

EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...