Equivariant twisted $R$-algebras via Thom spectra

Abstract

For a $C_2$-commutative ring spectrum $R$, a twisted $R$-algebra is an $R$-module with a multiplication whose order is switched by the $C_2$-action. In this paper, we construct various quotients of $R$ as twisted $R$-algebras, when $R$ is an even real commutative ring spectrum. These are constructed as Thom spectra of maps out of suitable $C_2$-actions on $S^1$ and $U(n)$. One such example is given by $K\mathbb{R}$ which is endowed with a twisted $K\mathbb{R}$-algebra structure. Other examples include quotients such as $M\mathbb{R}/(2,x_1,\dots, x_{n-1})$ over the real bordism spectrum $M\mathbb{R}$, and the real $2$-periodic Morava $K$-theories as modules over the real Morava $E$-theory spectra. In the context of twisted $R$-algebras, one may consider the real topological Hochschild homology, and for Thom spectra, one has a nice formula again as a Thom spectrum. We use this to obtain computations for the real topological Hochschild homology of $K\mathbb{R}/2$ as a twisted $K\mathbb{R}$-algebra. The computation also involves a splitting of the units spectrum $gl_1K\mathbb{R}$, which is an analogue of the classical splitting of the units of $K$-theory.