Topology Seminar

An Orientation Map for Higher Real E-theory

Hood Chatham, MIT

Thursday, October 31st, 2019
4:50 PM - 5:50 PM
Hylan 1106A

The real -theory spectrum is “almost complex oriented’’. Here are a collection of properties that demonstrate this:

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

(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas has a small Hurewicz image – it detects and .

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

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

Higher real theory is an odd primary analogue of . At , is closely related to . is defined as the 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 and prove that for any complex vector bundle the bundles and are complex oriented.

Event contact: vlorman at ur dot rochester dot edu