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A simple universal property of Thom ring spectra. (English) Zbl 1417.55007
The authors study the Thom spectrum of an \(\mathbb{E}_n\)-map (\(n \geq 0\)) using the model of Thom spectra in the \(\infty\)-category framework provided by [M. Ando et al., J. Topol. 7, No. 3, 869–893 (2014; Zbl 1312.55011)]. Recall that, if \(R\) is an \(\mathbb{E}_{n+1}\)-ring spectrum, the \(\infty\)-category \(\mathrm{Mod}_R\) of left \(R\)-modules is an \(\mathbb{E}_n\)-monoidal category; a local system of invertible \(R\)-modules on a space \(X\) is a map \(f : X \rightarrow \mathrm{Pic}(R)\) to the \(\infty\)-groupoid of invertible \(R\)-modules and the associated Thom spectrum is \(Mf := \mathrm{colim} (X \rightarrow \mathrm{Pic}(R) \rightarrow \mathrm{Mod}_R)\).
If \(X\) is an \(\mathbb{E}_n\)-space and \(f\) an \(\mathbb{E}_n\)-map, then \(Mf\) is an \(\mathbb{E}_n\) \(R\)-algebra. The authors give the following elegant characterization of the Thom spectrum \(Mf\): for an \(\mathbb{E}_n\) \(R\)-algebra \(A\), the mapping space \(\mathrm{Map}_{\mathrm{Alg}_R^{\mathbb{E}_n}} (Mf, A)\) is equivalent to the space of \(\mathbb{E}_n\)-lifts \(X \rightarrow \mathrm{Pic} (R)_{\downarrow A}\) of \(f\), where \(\mathrm{Pic} (R)_{\downarrow A} \) is the appropriate over-category. This is proved as a consequence of a general result in the \(\infty\)-categorical setting of algebras over an \(\infty\)-operad.
For \(A\) an \(\mathbb{E}_{n+1}\)-ring spectrum under \(R\), the authors revisit and develop the theory of \(A\)-orientations of the \(\mathbb{E}_n\)-map \(f\), for instance establishing the Thom isomorphism in this context. When \(f\) is an \(n\)-fold loop map, the space of \(A\)-orientations of \(f\) is shown to be equivalent to the space of \(\mathbb{E}_n\) \(R\)-algebra morphisms from \(Mf\) to \(A\). If follows, for example, that \(Mf\) is canonically \(\mathbb{E}_{n-1}\) \(Mf\)-orientable if \(n>0\).
They also generalize Szymik’s notion of characteristic [M. Szymik, Algebr. Geom. Topol. 14, No. 6, 3717–3743 (2014; Zbl 1311.55014)], giving the definition of the versal \(R\)-algebra \(R //_{\mathbb{E}_n} \chi\) associated to a characteristic \(\chi \in \pi_k R\) (\(k \geq 0\)). Given a map \(f : S^{k+1} \rightarrow BGL_1 (R)\) to the component of \(\mathrm{Pic} (R)\) containing \(R\), the authors associate a characteristic \(\chi (f)\) (the associated \(R\)-module map \(\Sigma^k R \rightarrow R\) if \(k>0\)) and they prove that there is an equivalence of \(\mathbb{E}_n\) \(R\)-algebras: \[ M \overline{f} \simeq R //_{\mathbb{E}_n} \chi (f), \] where \(\overline{f}\) is the \(n\)-fold loop map associated to \(f\).
As one application of their results, the authors recover the Hopkins-Mahowald realization of the Eilenberg-MacLane spectra \(H \mathbb{F}_p \) and \(H \mathbb{Z}\) as \(\mathbb{E}_2\)-ring spectra. This includes the fact that \(H \mathbb{F}_p\) is equivalent as an \(\mathbb{E}_2\)-ring spectrum to the versal characteristic \(p\) \(\mathbb{E}_2\)-algebra \(S^0 /\!/ _{\mathbb{E}_2} p\).

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55P42 Stable homotopy theory, spectra
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P48 Loop space machines and operads in algebraic topology
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