Timeline for Transfer of E-infinity algebra structures
Current License: CC BY-SA 4.0
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| Jan 21, 2022 at 21:45 | comment | added | J Cameron | @TylerLawson That's really interesting, thank you! I didn't know a strictly commutative object could have an interesting $E_{\infty}$ structure, but I suppose I shouldn't be surprised since the same thing happens in the $A_{\infty}$ case as you say. Do you know if anyone has done computations of the interesting $E_{\infty}$ structure for $H^*(X)$ for particular spaces? I assume that this is more information than just the Steenrod algebra structure? Also, do you know what you get if you transfer the trivial $E_{\infty}$ structure on $H^*(X)$ to $C^*(X,k)$? | |
| Jan 21, 2022 at 21:18 | history | edited | YCor |
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| Jan 21, 2022 at 20:13 | comment | added | Tyler Lawson | So $H^*(X;k)$ can get two $E_\infty$ structures. One is transferred from $C^*(X;k)$. The other is the trivial one, coming from the fact that $H^*(X;k)$ is a graded-commutative ring. The issue (which one also finds in the $A_\infty$-vs-associative case) is that these two aren't equivalent $E_\infty$ structures even though they have the same binary product. The Steenrod squares can be defined in terms of operations (extra data, beyond the product) that vanish for the second structure but not the first. | |
| Jan 21, 2022 at 19:17 | history | asked | J Cameron | CC BY-SA 4.0 |