BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jeff Carlson (Imperial College London) DTSTART:20211117T204500Z DTEND:20211117T220000Z DTSTAMP:20260423T022713Z UID:rts/3 DESCRIPTION:Title: Prod ucts on Tor\, homogeneous spaces\, and Borel cohomology\nby Jeff Carls on (Imperial College London) as part of Rochester topology seminar\n\nLect ure held in Hylan 1106A.\n\nAbstract\nThe Eilenberg-Moore spectral sequenc e converges from the classical Tor of a span of cohomology rings to the di fferential Tor of a span of cochain algebras (which is the cohomology of t he homotopy pullback). These are both rings\, the first classically and th e second as a corollary of the Eilenberg-Zilber theorem. \n\nOne might wel l ask when a more general differential Tor of DGAs admits a ring structure \, though apparently no one did. We will show that when the DGAs in questi on admit a certain sort of $E_3$-algebra structure\, Tor is a commutative graded algebra. \n\nWe have not done this out of an innocent interest in h omotopy-commutative algebras. In 1960s and '70s there was a flurry of acti vity developing A-infinity-algebraic techniques with an aim toward computi ng the Eilenberg–Moore spectral sequence (for example\, of a loop space or homogeneous space). Arguably the most powerful result this program prod uced was the 1974 theorem of Munkholm that the sequence collapses when the three input spaces have polynomial cohomology over a given principal idea l domain\, which however only gives the story on cohomology groups. Our re sult shows that Munkholm's map is in fact an isomorphism of rings. \n\nThe proof hinges on homotopy properties of the (1-)category of augmented DGAs . This work is all joint with several large commutative diagrams\, who sho uld be considered the true authors.\n\nZoom meeting ID: 988 2359 9895 pass code: 553391\n LOCATION:https://researchseminars.org/talk/rts/3/ END:VEVENT END:VCALENDAR