# nLab MO

Contents

### Context

#### Cobordism theory

Concepts of cobordism theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

relative bordism theories:

algebraic:

# 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 (Kochman 96, theorem 3.7.6)

The corresponding statement for MU is considerably more subtle, see Milnor-Quillen theorem on MU.

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

relative bordism theories:

algebraic:

## References

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

• Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)

Textbook accounts:

Review:

• Cary Malkiewich, section 2 of: Unoriented cobordism and $M O$, 2011 (pdf)

• Branko Juran, Thom spaces and the Oriented Cobordism Ring, 2020 (pdf, pdf)

Discussion of MO-bordism with MSO-boundaries:

• G. E. Mitchell, Bordism of Manifolds with Oriented Boundaries, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 (doi:10.2307/2040234)

In the incarnation of $MO$ as a symmetric spectrum:

In the incarnation as an orthogonal spectrum (in fact as an equivariant spectrum in global equivariant stable homotopy theory):

Last revised on March 9, 2021 at 07:28:02. See the history of this page for a list of all contributions to it.