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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Infinite loop spaces and positive scalar curvature
- Oscar Randal-Williams\, Cambridge
DTSTART;TZID=Europe/London:20131016T160000
DTEND;TZID=Europe/London:20131016T170000
UID:TALK46416AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/46416
DESCRIPTION:It is well known that there are topological obstru
ctions to a manifold $M$ admitting a Riemannian me
tric of everywhere positive scalar curvature (psc)
: if $M$ is Spin and admits a psc metric\, the Lic
hnerowiczâ€“WeitzenbĂ¶ck formula implies that the Dir
ac operator of $M$ is invertible\, so the vanishin
g of the $\\hat{A}$ genus is a necessary topologic
al condition for such a manifold to admit a psc me
tric. If $M$ is simply-connected as well as Spin\,
then deep work of Gromov--Lawson\, Schoen--Yau\,
and Stolz implies that the vanishing of (a small r
efinement of) the $\\hat{A}$ genus is a sufficient
condition for admitting a psc metric. For non-sim
ply-connected manifolds\, sufficient conditions fo
r a manifold to admit a psc metric are not yet und
erstood\, and are a topic of much current research
.\n\nI will discuss a related but somewhat differe
nt problem: if $M$ does admit a psc metric\, what
is the topology of the space $\\mathcal{R}^+(M)$ o
f all psc metrics on it? Recent work of V. Chernys
h and M. Walsh shows that this problem is unchange
d when modifying $M$ by certain surgeries\, and I
will explain how this can be used along with work
of Galatius and the speaker to show that the algeb
raic topology of $\\mathcal{R}^+(M)$ for $M$ of d
imension at least 6 is "as complicated as can poss
ibly be detected by index-theory". This is joint w
ork with Boris Botvinnik and Johannes Ebert.\n
LOCATION:MR13
CONTACT:Ivan Smith
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