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**The stable homotopy category is rigid.**
*(English)*
Zbl 1151.55007

The beautiful rigidity theorem asserts that if the homotopy category of a stable model category \(\mathcal C\) and that of spectra are equivalent as triangulated categories, then there exists a Quillen equivalence between \(\mathcal C\) and the model category of spectra.

If \(\Phi\) is such an equivalence of triangulated categories, the author chooses a fibrant object \(X\) in \(\mathcal C\) which is isomorphic to \(\Phi({\mathbf S}^0)\) in the homotopy category and proves that the functor \(- \wedge X\) and its right adjoint form a Quillen equivalence. For this it suffices to show that the composite \(F=\Phi^{-1} \circ (- \wedge^L X)\) is an equivalence on the homotopy category of spectra. The first step is a reduction to \(p\)-local spectra. Because the images under \(F\) of the Hopf maps are non-trivial and homotopy groups of spheres are generated under higher order Toda brackets by the Hopf maps and \(\alpha_1\), it is enough to see that the image of the map \(\alpha_1: {\mathbf S}^{2p-3} \rightarrow {\mathbf S}^{0}\) is non-trivial (this argument relies on the author’s earlier work [Adv. Math. 164, No.1, 24-40 (2001; Zbl 0992.55019)]).

The whole game is thus to find a contradiction if \(F(\alpha_1)\) were trivial, which is done via the theory of coherent actions of Moore spaces set up in the first sections. Let \(M\) be the Moore space \(M(\mathbb Z/p, 2)\) and \(D_i M = M^{\wedge i} \bigwedge_{\Sigma_i} E\Sigma_i^+\) be its \(i\)-th extended power. Roughly speaking a \(k\)-coherent action of \(M\) on \(X\) – and suspensions – consists in strictly associative and unital “multiplications” \(D_i M \wedge X_{(j)} \rightarrow X_{(i+j)}\) for \(i+j \leq k\), where the homotopical flexibility comes from the fact that \(X_{(j)}\) is only required to be homotopy equivalent to \(\Sigma^{2j-2} X\). The Moore space acts on itself in a \((p-1)\)-coherent way, but not in a \(p\)-coherent way due to the non-triviality of \(\alpha_1\). This also yields a \((p-1)\)-coherent action of \(M\) on \(M \wedge Y\) for any spectrum \(Y\).

Now, if \(F(\alpha_1)\) were trivial, the spectrum \(M \wedge F({\mathbf S}^{-2})\) would inherit a \(p\)-coherent action. This is the starting point of a clever iterative construction of spectra with certain prescribed non-trivial Steenrod operations which end up contradicting an Adem relation. What makes the construction possible is the compatibility with the triangulated structure, and in particular with the multiplication by \(p\) on a sphere (the homotopy cofiber of which is a Moore space). It is quite amazing that this little forces in the end all the higher structure to agree.

If \(\Phi\) is such an equivalence of triangulated categories, the author chooses a fibrant object \(X\) in \(\mathcal C\) which is isomorphic to \(\Phi({\mathbf S}^0)\) in the homotopy category and proves that the functor \(- \wedge X\) and its right adjoint form a Quillen equivalence. For this it suffices to show that the composite \(F=\Phi^{-1} \circ (- \wedge^L X)\) is an equivalence on the homotopy category of spectra. The first step is a reduction to \(p\)-local spectra. Because the images under \(F\) of the Hopf maps are non-trivial and homotopy groups of spheres are generated under higher order Toda brackets by the Hopf maps and \(\alpha_1\), it is enough to see that the image of the map \(\alpha_1: {\mathbf S}^{2p-3} \rightarrow {\mathbf S}^{0}\) is non-trivial (this argument relies on the author’s earlier work [Adv. Math. 164, No.1, 24-40 (2001; Zbl 0992.55019)]).

The whole game is thus to find a contradiction if \(F(\alpha_1)\) were trivial, which is done via the theory of coherent actions of Moore spaces set up in the first sections. Let \(M\) be the Moore space \(M(\mathbb Z/p, 2)\) and \(D_i M = M^{\wedge i} \bigwedge_{\Sigma_i} E\Sigma_i^+\) be its \(i\)-th extended power. Roughly speaking a \(k\)-coherent action of \(M\) on \(X\) – and suspensions – consists in strictly associative and unital “multiplications” \(D_i M \wedge X_{(j)} \rightarrow X_{(i+j)}\) for \(i+j \leq k\), where the homotopical flexibility comes from the fact that \(X_{(j)}\) is only required to be homotopy equivalent to \(\Sigma^{2j-2} X\). The Moore space acts on itself in a \((p-1)\)-coherent way, but not in a \(p\)-coherent way due to the non-triviality of \(\alpha_1\). This also yields a \((p-1)\)-coherent action of \(M\) on \(M \wedge Y\) for any spectrum \(Y\).

Now, if \(F(\alpha_1)\) were trivial, the spectrum \(M \wedge F({\mathbf S}^{-2})\) would inherit a \(p\)-coherent action. This is the starting point of a clever iterative construction of spectra with certain prescribed non-trivial Steenrod operations which end up contradicting an Adem relation. What makes the construction possible is the compatibility with the triangulated structure, and in particular with the multiplication by \(p\) on a sphere (the homotopy cofiber of which is a Moore space). It is quite amazing that this little forces in the end all the higher structure to agree.

Reviewer: Jérôme Scherer (Bellaterra)

### MSC:

55P42 | Stable homotopy theory, spectra |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55Q45 | Stable homotopy of spheres |

55S10 | Steenrod algebra |

18G55 | Nonabelian homotopical algebra (MSC2010) |