Contents

Idea

The universal Thom spectrum (see there for more) of the orthogonal group. (…) Abstractly, this is the homotopy colimit of the J-homomorphism in Spectra:

$MO = \underset{\rightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)$

Properties

Thom’s theorem on $M O$

By Thom's theorem the stable homotopy groups of $M O$ form the bordism ring of unoriented manifolds

$\pi_\bullet(M O) \simeq \Omega^O_\bullet \,.$

Moreover, this is the polynomial algebra

$\pi_\bullet(M O) \simeq (\mathbb{Z}/2\mathbb{Z})[ x_n \;|\; n \in \mathbb{N}, \,n \geq 2, \, n \neq 2^t-1] \,.$

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.

References

• René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86

Review includes

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.