# nLab bordism ring

Contents

## Theorems

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

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$012345678$\geq 9$
$\Omega^{SO}_k$$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$00$\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

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

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:

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:

Lecture notes:

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

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

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)