Two-primary algebraic \(K\)-theory of pointed spaces
In this impressive paper the author computes the \(2\)-primary homotopy type of Waldhausen's algebraic K-theory spectrum of a point \(A(*)\). This has far-reaching consequences; for instance, \(A(*)\) splits into two pieces, one of which is the sphere spectrum and the other is the smooth Whitehead spectrum \(\text{Wh}^{\text{Diff}}(*)\). The Whitehead spectrum in turn gives information about the group of diffeomorphisms of high-dimensional discs. To get optimal results in this direction, one needs to understand the involutive refinement of the results of the paper under review, and the author announces calculations to come in a later paper. The mathematics used in this paper is highly varied. Key ingredients are 1) Voevodsky's proof of the Milnor conjecture, and the proof of the Lichtenbaum-Quillen conjecture for the integers at two due to \textit{J. Rognes} and \textit{C. Weibel} [J. Am. Math. Soc. 13, No. 1, 1-54 (2000; Zbl 0934.19001)]; 2) the author's calculation [\textit{J. Rognes}, J. Pure Appl. Algebra 134, No. 3, 287-326 (1999; Zbl 0929.19004)] of topological cyclic homology of the integers at \(2\); 3) \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen}'s analysis of the topological cyclic homology of spaces [Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)] (the author provides a missing part in the argument), and the reviewer's result [\textit{B. I. Dundas}, Acta Math. 179, No. 2, 223-242 (1997; Zbl 0913.19002)] relating K-theory and topological cyclic homology through linearization; 4) an analysis of the mod \(2\) cohomology of the Thom spectrum \({\mathbb C}P^{\infty}_{-1}\) of minus the canonical line bundle over \({\mathbb C}P^{\infty}\). In what follows all spectra will tacitly be completed at \(2\). The calculation is performed by computing the mod \(2\) cohomology of \(A(*)\) as a module over the mod \(2\) Steenrod algebra. This is done by using the results in point 3) to convert the problem into a problem in topological cyclic homology. In effect, the author fits the Whitehead spectrum into a cofiber sequence \[ \mathbb C P^{\infty}_{-1}\to \text{hofib}(trc)\to\text{Wh}^{\text{Diff}}(*) \] where \(\text{hofib}(trc)\) is the homotopy fiber of the cyclotomic trace \(trc\colon K(\mathbb Z)\to TC(\mathbb Z)\). This term is taken care of by point 1) and 2) above, and the remaining problem is to analyze \({\mathbb C}P^{\infty}_{-1}\). The authors analyze the \(2\)-primary homotopy type of \({\mathbb C} P^{\infty}_{-1}\), calculating the homotopy groups in dimensions less than \(21\), leading to a calculation of the \(2\)-primary homotopy groups of the Whitehead spectrum in dimension less than \(19\). The cohomological calculations are, however, easier, and the author gives the mod \(2\) cohomology of \({\mathbb C} P^{\infty}_{-1}\) as a module over the mod \(2\) Steenrod algebra. There are various interesting spin-offs. For instance, the author shows that the linearization map \(A(*)\to K(\mathbb Z)\) induces zero in mod \(2\) spectrum cohomology in positive dimensions, and the space level Hatcher-Waldhausen map \(G/O\to\Omega Wh^{\text{Diff}}(*)\) does not admit a fourfold delooping.
scientific article; zbMATH DE number 1801882