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Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group. (This simplicial group has the homotopy type of the based loop space of $S^n$, so its homotopy groups compute the homotopy groups of $S^n$ shifted by one degree.)

For each simplicial degree $k$ define $N_k(S^n) \subset G_k(S^n)$ to be the intersection of the kernels of all but the last face maps $d_i: G_k(S^n) \to G_{k-1}(S^n)$. Then the last face map is a homomorphism $d_k: N_k(S^n) \to N_{k-1}(S^n)$. Moreover, the simplicial identities show $d_kd_{k+1}$ has constant value $1$, so we get a non-abelian chain complex of free groups. Its homology,'' "homology," by a result of Kan, computes $\pi_*(S^n)$.

To get an algorithm , for computing this homology, recall that the proof of the Nielsen-Schrier theorem gives a system of generators for the subgroup of any free group. So we obtain a system of generators for $N_k(S^n)$ as well as a system of generators for the image of $d_{k+1}$. So in principle we obtain a method for computing the homotopy groups of spheres.

In Kan's paper, $\pi_3(S^2)$ is computed in this way, and it takes several pages––so it's not a very good algorithm!

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Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group. (This simplicial group has the homotopy type of the based loop space of $S^n$, so its homotopy groups compute the homotopy groups of $S^n$ shifted by one degree.)

For each simplicial degree $k$ define $N_k(S^n) \subset G_k(S^n)$ to be the intersection of the kernels of all but the last face maps $d_i: G_k(S^n) \to G_{k-1}(S^n)$. Then the last face map is a homomorphism $d_k: N_k(S^n) \to N_{k-1}(S^n)$. Moreover, the simplicial identities show $d_{k+1}d_k = 1$, d_kd_{k+1}$has constant value$1$, so we get a non-abelian chain complex of free groups. Its homology,'' by a result of Kan computes$\pi_*(S^n)$. To get an algorithm, for computing this homology, recall that the proof of the Nielsen-Schrier theorem gives a system of generators for the subgroup of any free group. So we obtain a system of generators for$N_k(S^n)$as well as a system of generators for the image of$d_{k+1}$. So in principle we obtain a method for computing the homotopy groups of spheres. In Kan's paper,$\pi_3(S^2)$is computed in this way, and it takes several pages––so it's not a very good algorithm! 1 Here's a very useless algorithm due to Kan: Let$G(S^n)$be the simplicial group that is Kan loop group of the$n$-sphere. In each simplicial degree, it is a free group. (This simplicial group has the homotopy type of the based loop space of$S^n$, so its homotopy groups compute the homotopy groups of$S^n$shifted by one degree.) For each simplicial degree$k$define$N_k(S^n) \subset G_k(S^n)$to be the intersection of the kernels of all but the last face maps$d_i: G_k(S^n) \to G_{k-1}(S^n)$. Then the last face map is a homomorphism$d_k: N_k(S^n) \to N_{k-1}(S^n)$. Moreover, the simplicial identities show$d_{k+1}d_k = 1$, so we get a chain complex of free groups. Its homology,'' by a result of Kan computes$\pi_*(S^n)$. To get an algorithm, for computing this homology, recall that the proof of the Nielsen-Schrier theorem gives a system of generators for the subgroup of any free group. So we obtain a system of generators for$N_k(S^n)$as well as a system of generators for the image of$d_{k+1}$. So in principle we obtain a method for computing the homotopy groups of spheres. In Kan's paper,$\pi_3(S^2)\$ is computed in this way, and it takes several pages––so it's not a very good algorithm!