## 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.

### 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)

Zbl 0992.55019
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