Timeline for Poincare duality and the $A_\infty$ structure on cohomology
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2017 at 23:19 | comment | added | Pedro | "It is also true that the C∞ multiplication on cochains is a complete invariant of the rational homotopy type of a simply connected space, but I don't know of a place where this is written down explicitly. " What about this paper by Kadeishivili? | |
| Dec 31, 2010 at 20:12 | comment | added | David Ben-Zvi | I don't know if asking this algebra to be a commutative algebra object in addition gives the right notion of commutative ∞-Frobenius algebra, which is presumably what is required here. My guess would be no, since the commutative algebra structure isn't really interacting with the Frobenius structure. Then again, Jacob recovers string topology exactly this way from cochains on a Poincare duality space.. | |
| Dec 31, 2010 at 20:05 | comment | added | David Ben-Zvi | One natural definition of (noncommutative) $\infty$-separable algebra that Jeffrey is alluding to is as a fully dualizable object in the symmetric monoidal $(\infty,2)$ category of algebras, bimodules and maps of bimodules associated to the symmetric monoidal $(\infty,1)$-category $k-mod$ of dg vector spaces or simplicial k-modules for a ring k (which Jacob denotes $Alg_{(1)}(k-mod)$). The Frobenius condition is replaced by asking for an $SO(2)$-invariant such object, i.e. a Calabi-Yau algebra. | |
| Dec 31, 2010 at 15:06 | comment | added | Mark Grant | Re "$C_{\infty}$ multiplication on cochains is a complete invariant of the rational homotopy type of a simply connected space...": Kadeishvili proved the stronger(?) statement that there is a $C_\infty$ structure on cohomology which determines rational homotopy type. See his paper in the Proceedings of the Postnikov memorial conference. | |
| Dec 24, 2010 at 20:15 | comment | added | Nathaniel Rounds | Would such an object have a multiplication? It sounds like you giving an infinity version of a vector space with duality, not an infinity version of an algebra with duality. | |
| Dec 24, 2010 at 11:47 | comment | added | Jeffrey Giansiracusa | I suppose an $\infty$-Frobenius object is meant to be a fully dualizable object in a symmetric monoidal $(\infty,1)$-category, to use Lurie-esque language. | |
| Dec 24, 2010 at 11:06 | comment | added | Jeffrey Giansiracusa | Hi Nathaniel, thanks for the answer and welcome to MO! | |
| Dec 23, 2010 at 20:02 | history | answered | Nathaniel Rounds | CC BY-SA 2.5 |