nLab MO

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.

Related concepts

• homology of MO

• orientation in generalized cohomology

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

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

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

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:

• Robert Stong, Chapter VI of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)

• Stanley Kochman, section 1.5 and section 3.7 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

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:

• Stefan Schwede, Example I.2.8 in Symmetric spectra, 2012 (pdf)

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

• Stefan Schwede, chapter V.4 of Global homotopy theory, 2015