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SUMMARY:Infinite loop spaces and positive scalar curvature - Oscar Randal-
 Williams\, Cambridge
DTSTART:20131016T150000Z
DTEND:20131016T160000Z
UID:TALK46417@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:It is well known that there are topological obstructions to a 
 manifold $M$ admitting a Riemannian metric of everywhere positive scalar c
 urvature (psc): if $M$ is Spin and admits a psc metric\, the Lichnerowicz
 –Weitzenböck formula implies that the Dirac operator of $M$ is invertib
 le\, so the vanishing of the $\\hat{A}$ genus is a necessary topological c
 ondition for such a manifold to admit a psc metric. If $M$ is simply-conne
 cted as well as Spin\, then deep work of Gromov--Lawson\, Schoen--Yau\, an
 d Stolz implies that the vanishing of (a small refinement of) the $\\hat{A
 }$ genus is a sufficient condition for admitting a psc metric. For non-sim
 ply-connected manifolds\, sufficient conditions for a manifold to admit a 
 psc metric are not yet understood\, and are a topic of much current resear
 ch.\n\nI will discuss a related but somewhat different problem: if $M$ doe
 s admit a psc metric\, what is the topology of the space $\\mathcal{R}(M)$
  of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows t
 hat this problem is unchanged when modifying $M$ by certain surgeries\, an
 d I will explain how this can be used along with work of Galatius and the 
 speaker to show that the algebraic topology of $\\mathcal{R}(M)$ for $M$  
 of dimension at least 6 is "as complicated as can possibly be detected by 
 index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.
 \n
LOCATION:MR13
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