[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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