On the homotopy type of L‐spectra of the integers

Abstract

We show that quadratic and symmetric normalL‐theory of the integers are related by Anderson duality and that both spectra split integrally into the normalL‐theory of the real numbers and a generalised Eilenberg–Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G/Top. Finally, we prove analogous results for the genuine L‐spectra recently devised for the study of Grothendieck–Witt theory.

The family signature theorem

“…Following Hebestreit, Land and Nikolaus [21], the symmetric L-theory spectrum of the integers may be described as follows: there is a fibration sequence dR −→ L s (Z) −→ L s (R), (5.1) where π * (L s (R)) = Z[x ±1 ] with |x| = 4 and π i (dR) is Z/2 if i ≡ 1 mod 4 and is zero otherwise. The homotopy groups in degrees 4i detect the signature, and those in degrees 4i + 1 detect the de Rham invariant.…”

L‐theory of C∗$C^*$‐algebras

“…By Anderson duality, this shows that one can recover the abelian group KO 4 (𝐵𝐺) from L * (𝐶 * 𝑟 𝐺). Indeed, there is a short exact sequence 0 ⟶ Ext 1 ℤ (KO −1 (𝐵𝐺), ℤ) ⟶ KO 4 (𝐵𝐺) ⟶ Hom ℤ (KO 0 (𝐵𝐺), ℤ) ⟶ 0, which splits noncanonically; see, for example, [21] for a review of Anderson duality and the fact that the Anderson dual of KO is given by Ω 4 KO.…”

“…One can show that the map 𝜓 2 (in both the real and the complex case) exists as a map of 𝔼 1algebras. To construct this, one can use that 𝓁(ℝ) and 𝓁(ℂ) are 2-locally the free 𝔼 1 -𝐻ℤ-algebra on a generator in degree 4 and 2, respectively, see also [21,Corollary 4.2]. Then, the map 𝜓 2 is constructed as to be an 𝐻ℤ-algebra map at prime 2.…”

“…Then we get (1)$$\mathrm{L}_0(C^*_rG) = \mathrm{KO}_0(BG)$$, (2)$$\mathrm{L}_3(C^*_rG) = \mathrm{KO}_{-1}(BG)$$. By Anderson duality, this shows that one can recover the abelian group $$\mathrm{KO}^4(BG)$$ from $$\mathrm{L}_*(C^*_rG)$$. Indeed, there is a short exact sequence $$$$\begin{equation*} 0 \longrightarrow \mathrm{Ext}^1_\mathbb {Z}(\mathrm{KO}_{-1}(BG),\mathbb {Z}) \longrightarrow \mathrm{KO}^4(BG) \longrightarrow \mathrm{Hom}_\mathbb {Z}(\mathrm{KO}_0(BG),\mathbb {Z}) \longrightarrow 0, \end{equation*}$$$$which splits noncanonically; see, for example, [21] for a review of Anderson duality and the fact that the Anderson dual of $$\mathrm{KO}$$ is given by $$\Omega ^4 \mathrm{KO}$$.…”

The family signature theorem

“…The symmetric L-theory spectrum. Following Hebestreit, Land, and Nikolaus [HLN21], the symmetric L-theory spectrum of the integers may be described as follows: there is a fibration sequence…”