December 5, 2019
11:30 am
-
12:30 pm
MW 154
Speaker: Hood Chatham (MIT)
Title: An Orientation Map for Higher Real E-theory
Abstract: The real $\mathrm{K}$-theory spectrum $\mathrm{KO}$ is "almost complex oriented''. Here are a collection of properties that demonstrate this:
(1) $\mathrm{KO}$ is the $C_2$ fixed points of a complex oriented cohomology theory $\mathrm{KU}$.
(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas $\mathrm{KO}$ has a small Hurewicz image -- it detects $\eta$ and $\eta^2$.
(3) Complex oriented cohomology theories receive a ring map from $\mathrm{MU}$. $\mathrm{KO}$ receives no ring map from $\mathrm{MU}$ but it receives one from $\mathrm{MSU}$.
(4) If $E$ is a complex orientable cohomology theory, every complex vector bundle $V$ is $E$-orientable. Not every complex vector bundle $V$ is $\mathrm{KO}$-orientable, but $V\oplus V$ and $V^{\otimes 2}$ are.
Higher real $E$ theory $\mathrm{EO}$ is an odd primary analogue of $\mathrm{KO}$. At $p=3$, $\mathrm{EO}$ is closely related to $\mathrm{TMF}$. $\mathrm{EO}$ 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 $\mathrm{EO}$ and prove that for any complex vector bundle $V$ the bundles $pV$ and $V^{\otimes p}$ are complex oriented.
