group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The universal Thom spectrum (see there for more) of the orthogonal group. (…) Abstractly, this is the homotopy colimit of the J-homomorphism in Spectra:
By Thom's theorem the stable homotopy groups of $M O$ form the bordism ring of unoriented manifolds
Moreover, this is the polynomial algebra
Due to (Thom 54). See for instance (Kochmann 96, theorem 3.7.6)
The corresponding statement for MU is considerably more subtle, see Milnor-Quillen theorem on MU.
Review includes
Stanley Kochmann, section 1.5 and section 3.7 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Cary Malkiewich, section 2 of Unoriented cobordism and $M O$, 2011 (pdf)
In the incarnation of $MO$ as a symmetric spectrum is discussed in
In the incarnation as an orthogonal spectrum (in fact as an equivariant spectrum in global equivariant stable homotopy theory):
Last revised on June 15, 2017 at 09:40:02. See the history of this page for a list of all contributions to it.