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
   • CommentTimeJan 22nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have copied the nice implication flow chart from Adams' original paper into the entry, [here](https://ncatlab.org/nlab/show/Hopf+invariant+one#RelationToHSpaceStructureAndNormedDivisionAlgebras)

   I have copied the nice implication flow chart from Adams’ original paper into the entry, here

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2016
   • (edited Jan 23rd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have been adding little bits and pieces to _[[Hopf invariant]]_ and to _[[Hopf invariant one]]_. It occurs to me that for the quaternionic case of $n = 4$, the Hopf-invariant-one theorem has a neat interpretation in terms of the story of M2/M5-brane charges ([[schreiber:The WZW term of the M5-brane|here]]): Namely the $L_\infty$-computation which shows that the M2/M5-charge rationally lands in the base $S^4_{rat}$ of the rational quaternionic Hopf fibration has two natural non-rational integration: on the one hand to the actual $S^4$, and on the other hand -- more "conservatively" -- to the homotopy fiber of the cup product on ordinary cohomology: $hofib(\mathbf{B}^4 \mathbb{Z} \stackrel{\cup}{\to} \mathbf{B}^8 \mathbb{Z}) $. It seems plausible that the former is the genuine case to be considered for M2/M5-branes, but the latter certainly also still captures important information, and so a natural question is how the two compare. Drawing the evident hexagonal diagram $$ \array{ && S^7 \\ & {}^{\mathllap{\phi}}\swarrow && \searrow \\ S^4 && && \ast \\ \downarrow && && \downarrow \\ \mathbf{B}^4 \mathbb{Z} && && \ast \\ & {}_{\mathllap{\cup}}\searrow && \swarrow \\ && \mathbf{B}^8 \mathbb{Z} } $$ (with $[S^4 \to \mathbf{B}^4 \mathbb{Z}] = 1$ and with unit homotopy filling the diagram) we get a canonical comparison map induced: $$ S^7 \longrightarrow fib(\cup) \,. $$ So the natural question is: is this comparison faithful? Does it for instance send unit M2-brane charge in degree-4 cohomotopy to unit M2-brane charge in the more ordinary cohomology with coefficients in $fib(\cup)$. Now, simply by factoring the above homotopy differently, there is a sibling of the above comparison map induced: $$ cofib(\phi) \longrightarrow \mathbf{B}^8 \mathbb{Z} \,. $$ But inspection of the proof of the Hopf-invariant-one theorem shows that it says that the map that takes the class of $S^4 \to \mathbf{B}^{4}\mathbb{Z}$ to the class of this sibling comparison map, indeed sends generators to generators! But passing back to the dual picture, this should exactly mean that our original comparison map $$ S^7 \to fib(\cup) $$ sends generators to generators. Which is what we would hope it does. For example, for a single black M2-brane with near-horizon geometry $$ AdS_4 \times S^7 \simeq_{homotopy} S^7 \,, $$ this says that its unit charge as measured in degree-4 cohomotopy $$ S^7 \to S^4 $$ is sent under the comparison map to unit charge in the more naive almost-ordinary cohomology $$ S^7 \to fib(\cup) \,. $$ In conclusion, it seems that the Hopf invariant one theorem may be read as providing a further nontrivial consistency on the statement that M2/M5-brane charge is, non-rationally, in cohomotopy.

   I have been adding little bits and pieces to Hopf invariant and to Hopf invariant one.

   It occurs to me that for the quaternionic case of $n = 4$, the Hopf-invariant-one theorem has a neat interpretation in terms of the story of M2/M5-brane charges (here):

   Namely the $L_\infty$-computation which shows that the M2/M5-charge rationally lands in the base $S^4_{rat}$ of the rational quaternionic Hopf fibration has two natural non-rational integration: on the one hand to the actual $S^4$, and on the other hand – more “conservatively” – to the homotopy fiber of the cup product on ordinary cohomology: $hofib(\mathbf{B}^4 \mathbb{Z} \stackrel{\cup}{\to} \mathbf{B}^8 \mathbb{Z})$.

   It seems plausible that the former is the genuine case to be considered for M2/M5-branes, but the latter certainly also still captures important information, and so a natural question is how the two compare.

   Drawing the evident hexagonal diagram

   $$\array{ && S^7 \\ & {}^{\mathllap{\phi}}\swarrow && \searrow \\ S^4 && && \ast \\ \downarrow && && \downarrow \\ \mathbf{B}^4 \mathbb{Z} && && \ast \\ & {}_{\mathllap{\cup}}\searrow && \swarrow \\ && \mathbf{B}^8 \mathbb{Z} }$$

   (with $[S^4 \to \mathbf{B}^4 \mathbb{Z}] = 1$ and with unit homotopy filling the diagram)

   we get a canonical comparison map induced:

   $$S^7 \longrightarrow fib(\cup) \,.$$

   So the natural question is: is this comparison faithful? Does it for instance send unit M2-brane charge in degree-4 cohomotopy to unit M2-brane charge in the more ordinary cohomology with coefficients in $fib(\cup)$.

   Now, simply by factoring the above homotopy differently, there is a sibling of the above comparison map induced:

   $$cofib(\phi) \longrightarrow \mathbf{B}^8 \mathbb{Z} \,.$$

   But inspection of the proof of the Hopf-invariant-one theorem shows that it says that the map that takes the class of

   $S^4 \to \mathbf{B}^{4}\mathbb{Z}$

   to the class of this sibling comparison map, indeed sends generators to generators!

   But passing back to the dual picture, this should exactly mean that our original comparison map

   $$S^7 \to fib(\cup)$$

   sends generators to generators. Which is what we would hope it does.

   For example, for a single black M2-brane with near-horizon geometry

   $$AdS_4 \times S^7 \simeq_{homotopy} S^7 \,,$$

   this says that its unit charge as measured in degree-4 cohomotopy

   $$S^7 \to S^4$$

   is sent under the comparison map to unit charge in the more naive almost-ordinary cohomology

   $$S^7 \to fib(\cup) \,.$$

   In conclusion, it seems that the Hopf invariant one theorem may be read as providing a further nontrivial consistency on the statement that M2/M5-brane charge is, non-rationally, in cohomotopy.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 24th 2016
   • (edited Jan 27th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have spelled out the possibly cryptic remark above with a few more diagrams for better illustration.

   I have spelled out the possibly cryptic remark above with a few more diagrams for better illustration.