manifolds and cobordisms
cobordism theory, Introduction
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
hence
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
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)
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)
which justifies calling $\Omega_\bullet^G$ both the “bordism ring” as well as the “cobordism ring”.
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
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.
By Thom's theorem, for any (B,f)-structure $\mathcal{B}$, there is an isomorphism (of commutative rings)
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
is equivalently the sphere spectrum. Hence in this case Thom's theorem states that there is an isomorphism
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.
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)
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
e.g. Connor-Floyd 62, theorem 14.2
Textbook accounts include
Lecture notes include
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 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)
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?:
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)
Last revised on May 18, 2018 at 06:32:57. See the history of this page for a list of all contributions to it.