nLab
bordism ring

Contents

Idea

The (co)bordism ring Ω *= n0Ω n\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 Ω * SO\Omega^{SO}_*,

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

etc.

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

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

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

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

  • multiplication of [f 1:Σ 1X][f_1 \colon \Sigma_1 \to X] with [f 2:Σ 2X][f_2 \colon \Sigma_2 \to X] is given by transversal intersection Σ 1 XΣ 2\Sigma_1 \cap_X \Sigma_2 over XX: perturb f 1f_1 such (f 1,f 2)(f_1',f_2) becomes a transversal maps and then form the pullback Σ 1× (f 1,f 2)Σ 2\Sigma_1 \times_{(f_1',f_2)} \Sigma_2 in Diff.

This product is graded in that it satisfies the dimension formula

(dimXdimΣ 1)+(dimXdimΣ 2)=dimXdim(Σ 1 XΣ 2) (dim X - dim \Sigma_1) + (dim X - dim \Sigma_2) = dim X - dim (\Sigma_1 \cap_X \Sigma_2)

hence

dim(Σ 1 XΣ 2)=(dimΣ 1+dimΣ 2)dimX. dim (\Sigma_1 \cap_X \Sigma_2 ) = (dim \Sigma_1 + dim \Sigma_2) - dim X \,.

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

The resulting functor

(X,A)Ω G(X,A) (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 MGM G (e.g. MO for G=OG=O or MU for G=UG = 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)

Ω GMG (*)pi (MG). \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 MGM G-generalized cohomology of the point (cobordism cohomology theory)

Ω GMG (*) \Omega_\bullet^G \simeq M G^\bullet(\ast)

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

Properties

Examples

Framed cobordism

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

Ω π (M) \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 =Fr\mathcal{B} = Fr framing structure, then

MFr𝕊 M Fr \simeq \mathbb{S}

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

Ω frπ (𝕊) \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 π (𝕊)\pi_\bullet(\mathbb{S}) this way, see for instance (Wang-Xu 10, section 2).

For instance

  • Ω 0 fr=\Omega^{fr}_0 = \mathbb{Z} because there are two kk-framings on a single point, corresponding to π 0(O(k)) 2\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;

  • Ω 1 fr= 2\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 π 1(O(k)) 2\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

kk0123456789\geq 9
Ω k SO\Omega^{SO}_k\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_200\mathbb{Z}\oplus \mathbb{Z}0\neq 0

see e.g. (ManifoldAtlas)

Proposition

For XX a CW-complex (for instance a manifold), then the oriented cobordism ring is expressed in terms of the ordinary homology H q(X,Ω pq SO)H_q(X,\Omega^{SO}_{p-q}) of XX with coefficients in the cobordism ring over the point, prop. 1, as

Ω p SO(X)= q=0 pH q(X,Ω pq SO)mododdtorsion. \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

References

Textbook accounts include

Lecture notes include

Details for framed cobordism are spelled out in

  • 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)

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 GG-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 /p\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)

Revised on May 31, 2016 13:42:40 by Urs Schreiber (131.220.184.222)