Timeline for Is the $H$-space structure on $S^7$ associative up to homotopy?
Current License: CC BY-SA 3.0
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| Dec 6, 2016 at 7:21 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
to distinguish between Steenrod operations and projective spaces
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| Dec 5, 2016 at 19:20 | comment | added | Prasit | What you need is $A_{\infty}$-structure (slightly weaker than strict associativity) to construct classifying space/bar complex. Stasheff showed that you can constructed an $n$-truncated bar-complex for an $H$-space which is $A_n$. In particular homotopy associativity is equivalent to $A_3$-structure. If $S^7$ admits $A_3$-structure then you construct the $3$-truncated bar-complex aka $\mathbb{OP}^{3}$, and that is all is required for the proof above. | |
| Dec 5, 2016 at 18:37 | comment | added | SashaP | Isn't the associativity itself required to construct the classifying space which turns out to have the same cohomology as $\mathbb{OP}^{\infty}$? (assuming I got the argument right) | |
| Dec 5, 2016 at 10:40 | comment | added | Drew Heard | The answer to your question seems to be no: link.springer.com/chapter/10.1007%2FBFb0069236#page-1 | |
| Dec 4, 2016 at 23:24 | history | answered | Prasit | CC BY-SA 3.0 |