# zbMATH — the first resource for mathematics

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
##### Keywords:
Thom spectrum; $$E_n$$ structure; orientation; characteristic
Full Text: