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February 11, 2020 | 2:00 p.m. - 3:00 p.m.
Category: Seminar
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Cost: Free

Speaker: Hood Chatham, MIT

Title: An Orientation Map for Height p-1 Real E theory

Abstract: The real K-theory spectrum KO is ``almost complex oriented''. Here are a collection of properties that demonstrate this:

(1) KO is the C_2 fixed points of a complex oriented cohomology theory KU.

(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas KO has a small Hurewicz image -- it only detects (the v_1^4 family on) eta and eta^2.

(3) Complex oriented cohomology theories receive a ring map from MU.  KO receives no ring map from MU but it receives one from MSU.

(4) If E is a complex orientable cohomology theory, every complex vector bundle V is E-orientable. Not every complex vector bundle V is KO-orientable, but V oplus V and V otimes V are.

Higher real E theory EO is an odd primary analogue of KO. At p=3, EO is closely related to TMF. EO is defined as the C_p fixed points of a complex oriented cohomology theory, and it has a small but nontrivial Hurewicz image, so it satisfies analogues of properties (1) and (2). I prove that it also satisfies analogues of properties (3) and (4). In particular, I produce a unital orientation map from a Thom spectrum to EO and prove that for any complex vector bundle V, the bundles pV and V^{otimes p} are EO oriented.

 

For more information about this event, please contact Department of Mathematics at 313-577-2479 or math@wayne.edu.