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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 1st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[[periodic ring spectrum]]_ and to _[[periodic cohomology theory]]_ a brief paragraph on looping/delooping periodicity on the $\infty$-modules, with a pointer to [this MO discussion](http://mathoverflow.net/q/46399/381)

   added to periodic ring spectrum and to periodic cohomology theory a brief paragraph on looping/delooping periodicity on the $\infty$-modules, with a pointer to this MO discussion

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat's an interesting example of a 6-periodic ring spectrum? And more generally for $(4k+2)$-periodicity, with $k \in \mathbb{N}$. Does anything spring to mind that would follow such a pattern in periodicity?

   What’s an interesting example of a 6-periodic ring spectrum?

   And more generally for $(4k+2)$-periodicity, with $k \in \mathbb{N}$. Does anything spring to mind that would follow such a pattern in periodicity?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 2nd 2014
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   Author: Todd_Trimble
   Format: MarkdownItexThis reminds me of Qiaochu's question [here](http://mathoverflow.net/questions/185834/are-there-periodicity-phenomena-in-manifold-topology-with-odd-period).

   This reminds me of Qiaochu’s question here.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2014
   • (edited Nov 2nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Todd. Yes, I had seen that question when looking around yesterday. Somehow what I am asking here should be more elementary. For instance where Qiaochu invokes the [ABS orientation](http://ncatlab.org/nlab/show/K-orientation#UniversalAtiyahBottShapiroOrientation) to explain 8-fold periodicity in manifolds from that of the $KO$-ring spectrum, here I'd be happy to just observe that $KO$ is 8-periodic. But I am really wondering about 6-periodicity. I may be wrong, but recenty it occurred to me that the 2-periodicity in elliptic cohomology is to be thought of as related to the 2+-dimensionality of Chern-Simons theory. Because, as in [this MO question](http://mathoverflow.net/q/185721/381) there is naturally an elliptic line bundle in codimension 1 and in codimension 3, one being the point restriction of the other, and for the one in codimension 3 to have a chance to reproduce the one in codimenesion 1 by transgression, we need 2-periodicity of its fibers. If that's the right perspective, then for 7d CS theory we'd expect to need 6-periodicity, and so that's why I was wondering if there are any candidates for that. In any case, of course one may alsways force anything to become $n$-periodic for any $n$. But I'd be happy to see ring spectra arising "in nature" that are 6-periodic.

   Thanks, Todd. Yes, I had seen that question when looking around yesterday.

   Somehow what I am asking here should be more elementary. For instance where Qiaochu invokes the ABS orientation to explain 8-fold periodicity in manifolds from that of the $KO$-ring spectrum, here I’d be happy to just observe that $KO$ is 8-periodic. But I am really wondering about 6-periodicity.

   I may be wrong, but recenty it occurred to me that the 2-periodicity in elliptic cohomology is to be thought of as related to the 2+-dimensionality of Chern-Simons theory. Because, as in this MO question there is naturally an elliptic line bundle in codimension 1 and in codimension 3, one being the point restriction of the other, and for the one in codimension 3 to have a chance to reproduce the one in codimenesion 1 by transgression, we need 2-periodicity of its fibers. If that’s the right perspective, then for 7d CS theory we’d expect to need 6-periodicity, and so that’s why I was wondering if there are any candidates for that.

   In any case, of course one may alsways force anything to become $n$-periodic for any $n$. But I’d be happy to see ring spectra arising “in nature” that are 6-periodic.