The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

Question

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the integers. There is another paper, 'The mod $2$ cohomology of the linear groups over the ring of integers', where the mod 2 cohomology of the the space $\BGL(\mathbb{Z})^+$ is computed. A paper by Hodgkin presents a computation of the $2$-local topological $K$-theory of $\BGL(\mathbb{Z})$ and $\BGL(\mathbb{Z})^+$. Inspired by this line of results, I have the following 2 questions:

1. Let $E$ be a (co)homology theory and $R$ be a ring. How does the understanding of the $E^\ast(\BGL(R))$ and $E^\ast(\BGL(R)^+)$ facilitate the understanding of the algebraic $K$-theory of $R$?
2. In the case of homology, how are $E_\ast(\BGL(R))$ and $E_\ast(\BGL(R)^+)$ related to the algebraic $K$-theory of $R$?

Answer 1

Let $R$ be a ring. $BGL(R)^+$ is homotopy equivalent to the $0$ component of $\Omega^\infty K(R)$, and it is stably equivalent to $BGL(R)$.

In particular, for a (co)homology theory $E$, understanding $E^*(BGL(R))$ is the same as understanding $E^*(BGL(R)^+)$, which in turn is the same as $E^*(\Omega^\infty K(R)_0$).

So in some sense your question amounts to "if $X$ is a spectrum and I understand $E^*(\Omega^\infty X)$, what does that tell me about $X$ ?". It's hard to answer in such generality, but there are many connections between the unstable homotopy type of a spectrum and the spectrum itself. For example, $T(n)$ and $K(n)$-localizations only depend on $\Omega^\infty$, so if you are interested in $L_{T(n)}(K(R))$, there is no loss in only remembering this.

There are also more basic relations, like the fact that $p$-completion commutes with $\Omega^\infty$ on $1$-connective spectra; but also $\Omega^\infty X$ is always a simple space, so that $H_*(\Omega^\infty(-);\mathbb Z)$ is conservative on connective spectra.