nLab
MO

Context

Cobordism theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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=lim(BOJBGL 1(𝕊)Spectra) MO = \underset{\rightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)

Properties

Thom’s theorem on MOM O

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

π (MO)Ω O. \pi_\bullet(M O) \simeq \Omega^O_\bullet \,.

Moreover, this is the polynomial algebra

π (MO)(/2)[x n|n,n2,n2 t1]. \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 MOMO 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.