Topological Hochschild homology of \(K/p\) as a \(K_p^\wedge\) module (Q1688699)

In order to calculate the topological Hochschild homology of \(K/p\), the mod \(p\) \(K\)-theory spectrum, the author uses generalized Thom spectra effectively. Let \(K_p^{\wedge}\) be the \(p\)-adic \(K\)-theory spectrum and let \(\zeta = 1-p : S^1 \to BLG_1(K_p^{\wedge})\) denote the map chosen via the isomorphism \(\pi_1(BGL_1(K_p^{\wedge}))\cong \pi_0(GL_1(K_p^{\wedge}))\cong {\mathbb Z}_p^{\times}\). Then we have an extension \(f : {\mathbb C}P^1 \to B^2GL_1 (K_p^{\wedge})\) of \(\Sigma(1-p)\) along the standard maps \(\Sigma S^1 \to {\mathbb C}P^1\) and \(\Sigma BGL_1 (K_p^{\wedge}) \to B^2GL_1 (K_p^{\wedge})\) to the classifying spaces. The main theorem of the paper under review asserts that for odd primes \(p\), \[ \pi_n(THH^{K_p^{\wedge}}(K/p, f)) = \begin{cases} ({\mathbb Z}/(p^\infty))^i & \text{if }n\text{ is even} \\ 0 & \text{if }n\text{ is odd}. \end{cases} \] Here \(i\) is an integer between \(1\) and \(p-1\) which is determined by the \(A_\infty\)-ring structure of \(K/p\simeq (S^1)^\zeta\) depending on the choice of \(f\) with \(\zeta \simeq \Omega f\). The key to proving the theorem is to regard the topological Hochschild homology as a Thom spectrum. In fact, one has a homotopy equivalence \(THH^{K_p^{\wedge}}(K/p, f) \simeq (L{\mathbb C}P^\infty)^{\widehat{f}}\), where \(\widehat{f}\) is the composite \(L{\mathbb C}P^\infty \overset{Lf}{\to} LB^2GL_1(K_p^{\wedge}) \simeq B^2GL_1(K_p^{\wedge}) \times BGL_1(K_p^{\wedge}) \overset{p_2}{\to} BGL_1(K_p^{\wedge})\). The Thom spectrum \((L{\mathbb C}P^\infty)^{\widehat{f}}\) fits into a pushout square of Thom spectra. Thus rewriting the Mayer-Vietoris sequence obtained by the pushout square, the author gives a long exact sequence of the form \[ \cdots \overset{}{\longrightarrow} {K_p^{\wedge}}_*({\mathbb C}P^\infty) \overset{u-1}{\longrightarrow} {K_p^{\wedge}}_*({\mathbb C}P^\infty) \overset{}{\longrightarrow} \pi_*((L{\mathbb C}P^\infty)^{\widehat{f}}) \overset{}{\longrightarrow} {K_p^{\wedge}}_{*-1}({\mathbb C}P^\infty) \overset{}{\longrightarrow} \cdots. \] Explicit consideration of the map \(u : \Sigma {\mathbb C}P^\infty_+ \to BGL_1(K_p^{\wedge})\) allows us to deduce the main theorem mentioned above.