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
   • CommentTimeApr 3rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded just a little at _[[KR-theory]]_ by giving it an actual Idea-paragraph and adding some more references.

   I have expanded just a little at KR-theory by giving it an actual Idea-paragraph and adding some more references.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a section _[Properties -- As induced from the derived moduli stack of tori](http://ncatlab.org/nlab/show/KR+cohomology+theory#InducedFromModuliStackOfTori)_ which currently reads as follows: *** The relation between $KU$, $KO$ and $KR$ naturally arises in [[chromatic homotopy theory]] as follows. Inside the [[moduli stack of formal group laws]] sits the _moduli stack of one dimensional tori_ $\mathcal{M}_{\mathbb{G}_m}$ ([#Lawson-Naumann 12, def. A.1, A.3](#LawsonNaumann12)). This is equivalent to the [[quotient stack]] of the point by the [[group of order 2]] $$ \mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2 $$ ([Lawson-Naumann 12, prop. A.4](#LawsonNaumann12)). Here the $\mathbb{Z}_2$-action is the inversion involution on [[abelian groups]]. Using the [[Goerss-Hopkins-Miller theorem]] this stack carries an [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{top}$ ([Lawson-Naumann 12, theorem A.5](#LawsonNaumann12)); and by the above equivalence this is a single [[E-∞ ring]] equipped with a $\mathbb{Z}_2$-[[∞-action]]. This is [[KU]] with its involution induced by [[complex conjugation]], hence essentially is $KR$. Accordingly, the [[global sections]] of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-[[homotopy fixed points]] of this action, hence is $KO$. This is further amplified in ([Mathew 13, section 3](#Mathew13)) and ([Mathew, section 2](#Mathew)). As suggested there and by the main ([Lawson-Naumann 12, theorem 1.2](#LawsonNaumann12)) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of [[tmf]] built as the global sections of the similarly derived [[moduli stack of elliptic curves]].

   added a section Properties – As induced from the derived moduli stack of tori which currently reads as follows:

   ---

   The relation between $KU$, $KO$ and $KR$ naturally arises in chromatic homotopy theory as follows.

   Inside the moduli stack of formal group laws sits the moduli stack of one dimensional tori $\mathcal{M}_{\mathbb{G}_m}$ (#Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2

   $$\mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2$$

   (Lawson-Naumann 12, prop. A.4). Here the $\mathbb{Z}_2$-action is the inversion involution on abelian groups.

   Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf $\mathcal{O}^{top}$ (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a $\mathbb{Z}_2$-∞-action. This is KU with its involution induced by complex conjugation, hence essentially is $KR$.

   Accordingly, the global sections of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-homotopy fixed points of this action, hence is $KO$. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).

   As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.

3. • CommentRowNumber3.
   • CommentAuthorColin Tan
   • CommentTimeMay 21st 2014
   • PermaLink
   Author: Colin Tan
   Format: MarkdownItexI am reading the entry on [KR cohomology theory](http://ncatlab.org/nlab/show/KR+cohomology+theory). Just at the end of the Idea section, there is a "remark on terminology 1" subsection. This subsection begins with "Hence X here is equipped with an involution by a diffeomorphism." I am not sure the "hence" is a consequence of which argument. Also the Idea section begins with introducing a real space as an "involution by a homeomorphism" so perhaps we are talking about a real manifold here?

   I am reading the entry on KR cohomology theory. Just at the end of the Idea section, there is a “remark on terminology 1” subsection. This subsection begins with “Hence X here is equipped with an involution by a diffeomorphism.” I am not sure the “hence” is a consequence of which argument. Also the Idea section begins with introducing a real space as an “involution by a homeomorphism” so perhaps we are talking about a real manifold here?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for catching that, this remark was moved a little out of its original context. I have slightly rephrased it to now read as follows: > An [[involution]] on a space by a [[homeomorphism]] (or [[diffeomorphism]]) as they appear in KR theory may be thought of as a "non-linear [[real structure]]", and therefore spaces equipped with such involutions are called "[[real spaces]]". Following this, $KR$-theory is usually pronounced "real K-theory". But **beware** that this terminology easily conflicts with or is confused with [[KO]]-theory. For disambiguation the latter might better be called "orthogonal K-theory". But on abstract grounds maybe $KR$-theory would best be just called $\mathbb{Z}_2$-equivariant complex K-theory. Regarding your last question: I am not sure if I understand what you are after. Maybe with the above rephrasing this is now clarified? Or maybe I am missing something.

   Thanks for catching that, this remark was moved a little out of its original context. I have slightly rephrased it to now read as follows:

   An involution on a space by a homeomorphism (or diffeomorphism) as they appear in KR theory may be thought of as a “non-linear real structure”, and therefore spaces equipped with such involutions are called “real spaces”. Following this, $KR$-theory is usually pronounced “real K-theory”. But beware that this terminology easily conflicts with or is confused with KO-theory. For disambiguation the latter might better be called “orthogonal K-theory”. But on abstract grounds maybe $KR$-theory would best be just called $\mathbb{Z}_2$-equivariant complex K-theory.

   Regarding your last question: I am not sure if I understand what you are after. Maybe with the above rephrasing this is now clarified? Or maybe I am missing something.

5. • CommentRowNumber5.
   • CommentAuthorColin Tan
   • CommentTimeJun 21st 2014
   • PermaLink
   Author: Colin Tan
   Format: TextThanks Urs: whatever I was confused about has been clarified with your rephrasing.

   Thanks Urs: whatever I was confused about has been clarified with your rephrasing.