[ALGTOP-L] Einfty question
Nicholas J. Kuhn
njk4x at virginia.edu
Fri Nov 19 16:02:40 EST 2010
If I got this right, here is an example related to Jim's query that
some of you may find amusing and/or disturbing:
The cochains on a circle C*(S^1) is "A_infty formal" but not "E_infty
formal".
In other words, C*(S^1) is equivalent to H*(S^1) as an A_infty
algebra, but not as an E_infty algebra. As a bonus property, rational
chains C*(S^1;Q) ARE E_infty formal.
To distinguish these as E_infty algebras one calculates their
(topological) Hochschild Homology (over Z).
HH(H*(S^1)) is a divided power algebra on a class "x": the subring of
Q[x] with basis x^k/k!.
HH(C*(S^1)) is the `numeric polynomial ring': the subring of Q[x] with
basis the polynomials `x choose k'.
This latter calculation is something that Mike Mandell and I had fun
figuring out once upon a time. (Note: If there are errors in this
message, they are mine and not his!)
Nick Kuhn
>> has anyone done the obstruction theory for extending an A_infty
>> quasi-iso A -> B between E_infty algebras to and homotopy
>>equivalence of
>> E_infty algebras?
>>
>> or having in mind the case where A is the deRham complex?
>> jim
>>
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