Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2017
   • (edited Jun 15th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am looking for a decent account of the homotopy ring spectrum structure on $KU$ with $KU_0 = BU \times \mathbb{Z}$ that would be self-contained for a reader with good point-set topology background, but not involving $E_\infty$ or model category theory. What I find in the literature is all sketchy, but maybe I am looking in the wrong places. First, a discussion of the H-space structure on $BU\times \mathbb{Z}$ in the first place I find on p. 205 (213 of 251) in _[[A Concise Course in Algebraic Topology]]_. But for the crucial step it there only says: > we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product Is there a reference that would spell this out? Next, for the proof of the homotopy ring spectrum structure on $KU$, the idea is indicated on the first page of James McLure, _$H_\infty$-ring spectra via space-level homotopy theory_ ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/BMMS8.pdf)) Is there a place where this would be spelled out in some detail?

   I am looking for a decent account of the homotopy ring spectrum structure on $KU$ with $KU_0 = BU \times \mathbb{Z}$ that would be self-contained for a reader with good point-set topology background, but not involving $E_\infty$ or model category theory.

   What I find in the literature is all sketchy, but maybe I am looking in the wrong places.

   First, a discussion of the H-space structure on $BU\times \mathbb{Z}$ in the first place I find on p. 205 (213 of 251) in A Concise Course in Algebraic Topology. But for the crucial step it there only says:

   we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product

   Is there a reference that would spell this out?

   Next, for the proof of the homotopy ring spectrum structure on $KU$, the idea is indicated on the first page of

   James McLure, $H_\infty$-ring spectra via space-level homotopy theory (pdf)

   Is there a place where this would be spelled out in some detail?