nLab bordism ring

Idea

The (co)bordism ring $\Omega_*=\oplus_{n\geq 0}\Omega_n$ is the graded ring whose

• degree $n$ elements are classes of $n$-dimensional smooth manifolds modulo cobordism;

• product operation is given by the Cartesian product of manifolds;

• addition operation is given by the disjoint union of manifolds.

Instead of bare manifolds one may consider manifolds with extra structure, such as orientation, spin structure, string structure, etc. and accordingly there is

• the oriented cobordism ring $\Omega^{SO}_*$,

• the spin cobordism ring $\Omega^{Spin}_*$,

etc.

In this general context the bare cobordism ring is also denoted $\Omega^O_*$ or $\Omega^{un}_*$, for emphasis.

A ring homomorphism out of the cobordism ring is a (multiplicative) genus.

More generally, for $X$ a fixed manifold there is a relative cobordism ring $\Omega_\bullet(X)$ whose

• elements are classes modulo cobordism over $X$ of manifolds equipped with continuous functions to $X$ (“singular manifolds”);

• multiplication of $[f_1 \colon \Sigma_1 \to X]$ with $[f_2 \colon \Sigma_2 \to X]$ is given by transversal intersection $\Sigma_1 \cap_X \Sigma_2$ over $X$: perturb $f_1$ such $(f_1',f_2)$ becomes a transversal maps and then form the pullback $\Sigma_1 \times_{(f_1',f_2)} \Sigma_2$ in Diff.

This product is graded in that it satisfies the dimension formula

$$(dim X - dim \Sigma_1) + (dim X - dim \Sigma_2) = dim X - dim (\Sigma_1 \cap_X \Sigma_2)$$

hence

$$dim (\Sigma_1 \cap_X \Sigma_2 ) = (dim \Sigma_1 + dim \Sigma_2) - dim X \,.$$

Still more generally, this may be considered for $\Sigma$ being manifolds with boundary. Then $\Omega(X,A)$ for $(X,A)$ a CW pair is the ring of cobordism classes, relative boundary, of singular manifolds $\Sigma \to X$ such that the boundary of $\Sigma$ lands in in $A$.

The resulting functor

$$(X,A) \mapsto \Omega^G_\bullet(X,A)$$

constitutes a generalized homology theory (see e.g. Buchstaber, II.8). Accordingly this is called bordism homology theory.

The spectrum that represents this under the Brown representability theorem is the universal Thom spectra $M G$ (e.g. MO for $G=O$ or MU for $G = U$), which canonically is a ring spectrum under Whitney sum of universal vector bundles. Accordingly the (co-bordism ring) itself is equivalently the bordism homology groups of the point, hence the stable homotopy groups of the Thom spectrum (this is Thom's theorem)

$$\Omega_\bullet^G \simeq \M G_\bullet(\ast) \simeq \pi_\bullet(M G) \,.$$

This remarkable relation between manifolds and stable homotopy theory is known as cobordism theory (or “Thom theory”).

On general grounds this is equivalently the $M G$-generalized cohomology of the point (cobordism cohomology theory)

$$\Omega_\bullet^G \simeq M G^\bullet(\ast)$$

which justifies calling $\Omega_\bullet^G$ both the “bordism ring” as well as the “cobordism ring”.

Properties

Relation to cohomotopy

Let $X$ be a smooth manifold of dimension $n \in \mathbb{N}$ and let $k \leq n$. Then the Pontryagin-Thom construction induces a bijection

$$[X, S^k] \overset{\simeq}{\longrightarrow} \Omega^{n-k}(X)$$

from the cohomotopy sets of $X$ to the cobordism group of $(n-k)$-dimensional submanifolds with normal framing up to normally framed cobordism.

In particular, the natural group structure on cobordism group (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets.

This is made explicit for instance in Kosinski 93, chapter IX.

Examples

Framed cobordism

By Thom's theorem, for any (B,f)-structure $\mathcal{B}$, there is an isomorphism (of commutative rings)

$$\Omega^{\mathcal{B}}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(M\mathcal{B})$$

from the cobordism ring of manifolds with stable normal $\mathcal{B}$-structure to the homotopy groups of the universal $\mathcal{B}$-Thom spectrum.

Now for $\mathcal{B} = Fr$ framing structure, then

$$M Fr \simeq \mathbb{S}$$

is equivalently the sphere spectrum. Hence in this case Thom's theorem states that there is an isomorphism

$$\Omega^{fr}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(\mathbb{S})$$

between the framed cobordism ring and the stable homotopy groups of spheres.

For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance (Wang-Xu 10, section 2).

For instance

• $\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;

• $\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.

Oriented cobordism

Proposition

The cobordism ring over the point for oriented manifolds starts out as

$k$	0	1	2	3	4	5	6	7	8	$\geq 9$
$\Omega^{SO}_k$	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	0	0	$\mathbb{Z}\oplus \mathbb{Z}$	$\neq 0$

see e.g. (ManifoldAtlas)

Proposition

For $X$ a CW-complex (for instance a manifold), then the oriented cobordism ring is expressed in terms of the ordinary homology $H_q(X,\Omega^{SO}_{p-q})$ of $X$ with coefficients in the cobordism ring over the point, prop. , as

$$\Omega_p^{SO}(X) = \oplus_{q = 0}^p H_q(X,\Omega_{p-q}^{SO}) \; mod\; odd \; torsion \,.$$

e.g. Connor-Floyd 62, theorem 14.2

Related concepts

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

Original articles:

• René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86 (digiz:GDZPPN002056259)

• John Milnor, On the cobordism ring ­$\Omega^\bullet$ and a complex analogue, Amer. J. Math. 82 (1960), 505–521 (jstor:2372970)

• Sergei Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).

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

• Daniel Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Mathematics 7 1 (1971) 29-56 [doi:10.1016/0001-8708(71)90041-7]

  (using Steenrod operations)

Textbook accounts:

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

• Stanley Kochmann, section 1.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

Lecture notes:

• Victor Buchstaber, Geometric cobordism theory (pdf)

• John Francis (notes by Owen Gwilliam), Topology of manifolds, Lecture 2: Cobordism (pdf)

• Gerald Höhn, Komplexe elliptische Geschlechter und $S^1$-äquivariante Kobordismustheorie (german) (pdf)

• Sander Kupers, Oriented bordism: Calculation and application (pdf)

Details for framed cobordism:

• Guozhen Wang, Zhouli Xu, section 2 of A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)

• Andrew Putman, Homotopy groups of spheres and low-dimensional topology (pdf)

The relation to cohomotopy is made explicit in

• Antoni Kosinski, chapter IX of Differential manifolds, Academic Press 1993 (pdf)

Further discussion of oriented cobordism includes

• Manifold Atlas, Oriented bordism

• P. E. Conner, E. E. Floyd, Differentiable periodic maps, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. (Euclid, pdf)

A historical review in the context of complex cobordism cohomology theory/Brown-Peterson theory is in

• Doug Ravenel, chapter 4, section 2 of Complex cobordism and stable homotopy groups of spheres

On fibered cobordism groups:

• Astey, Greenberg, Micha, Pastor, Some fibered cobordisms groups are not finitely generated (pdf)

Discussion of the $G$-equivariant complex coborism ring includes

• G. Comezana and Peter May, A completion theorem in complex cobordism, in Equivariant Homotopy and Cohomology Theory, CBMS Regional conference series in Mathematics, American Mathematical Society Publications, Volume 91, Providence, 1996.

• Igor Kriz, The $\mathbb{Z}/p$–equivariant complex cobordism ring, from: “Homotopy invariant algebraic structures (Baltimore, MD, 1998)”, Amer. Math. Soc. Providence, RI (1999) 217–223

• Neil Strickland, Complex cobordism of involutions, Geom. Topol. 5 (2001) 335-345 (arXiv:math/0105020)

• William Abrams, Equivariant complex cobordism, 2013 (pdf)