nLab
cobordism cohomology theory

Contents

Contents

Idea

In algebraic topology, a Whitehead-generalized cohomology theory represented by a Thom spectrum is called a cobordism cohomology theory (Atiyah 61), in duality with the corresponding generalized homology theory called bordism homology theory.

In both cases, a version of the Pontryagin-Thom construction identifies the (co)homology classes of these (co)homology theories with bordism-equivalence classes of manifolds (carrying some given extra structure), whence the name. For bordism homology theory this was understood since the very inception of the subject (Thom 54), while for cobordism cohomology theory this identification is made explicit in Atiyah 61, Sec. 3, Quillen 71 (relying on results from Thom 54 nonetheless), see below at Geometric model via cobordism classes.

Accordingly, cobordism cohomology theories are fundamental concepts of bordism theory in differential topology. But in addition they turn out to play a special role in the more abstract stable homotopy theory of complex oriented cohomology theories (with its variants such as quaternionic-oriented theories) and in the resulting chromatic homotopy theory, see for instance the universal complex orientation on MU. This way, cobordism cohomology embodies a remarkable confluence of the differential topology of smooth manifolds with deep issues in abstract homotopy theory.

There are many different flavours of cobordism cohomology theories (see the list of Examples below), depending on the tangential structure ff encoded in the representing Thom spectrum MfM f. Among the most commonly considered versions are these:

Geometric model via cobordism classes

We discuss a geometric model for the cobordism cohomology theory, due to Quillen 71, Section 1. We concentrate on the complex case, corresponding to the Thom spectrum MU:

Proposition

For a smooth manifold XX, the cobordism cohomology group MU q(X)[Σ X +,Σ qMU]\mathrm{M} \mathrm{U}^q(X) \;\coloneqq\; [\Sigma^\infty X_+, \Sigma^q MU] is equivalently the set of cobordism classes of proper complex-oriented maps f:ZXf \colon Z \to X of codimension qq.

(Quillen 71, Prop. 1.2)

This uses the following definitions:

Definition

(complex-oriented maps)

Let f:ZXf \colon Z \to X be a smooth map.

If the relative codimension of ff is even at all points of ZZ, then a complex orientation is an equivalence class of factorizations of ff in the form

pi:ZEX, p \circ i \;\colon\; Z \longrightarrow E \longrightarrow X \,,

where p:EXp\colon E\to X is a complex vector bundle and i:ZEi \colon Z\to E is an embedding equipped with a complex structure on its normal bundle.

Two such factorizations (i,p)(i,p) and (i,p)(i',p') are regarded as equivalent if there is another factorization (i,p)(i'',p'') together with embeddings of complex vector bundles EEE\to E' and EEE\to E'' and a homotopy i:X×[0,1]E×[0,1]i''\colon X\times[0,1]\to E''\times [0,1] over [0,1][0,1] equipped with a complex structure on its normal bundle that restricts to the corresponding complex structures on X×{0}X \times \{0\} and X×{1}X \times \{1\}.

Definition

(cobordism classes of maps)

Here two proper complex-oriented maps f i:Z iXf_i \colon Z_i \to X are called cobordant if there is a proper complex-oriented map b:WX×Rb\colon W\to X\times\mathbf{R} such that X×{0}X\times\{0\} and X×{1}X\times\{1\} are transversal to bb and pulling back bb to these submanifolds yields f 0f_0 and f 1f_1.

(Quillen 71, p. 31)

Examples

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:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:


chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References

General

Original articles introducing cobordism as a Whitehead-generalized cohomology theory:

Early survey:

Textbook accounts:

The twisted and equivariant versions:

Pontrjagin-Thom construction

Pontrjagin’s construction

General

The Pontryagin theorem, i.e. the unstable and framed version of the Pontrjagin-Thom construction, identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge in unstable Borsuk-Spanier Cohomotopy sets, is due to:

(both available in English translation in Gamkrelidze 86),

as presented more comprehensively in:

The Pontrjagin theorem must have been known to Pontrjagin at least by 1936, when he announced the computation of the second stem of homotopy groups of spheres:

  • Lev Pontrjagin, Sur les transformations des sphères en sphères (pdf) in: Comptes Rendus du Congrès International des Mathématiques – Oslo 1936 (pdf)

Review:

Discussion of the early history:

Twisted/equivariant generalizations

The (fairly straightforward) generalization of the Pontrjagin theorem to the twisted Pontrjagin theorem, identifying twisted Cohomotopy with cobordism classes of normally twisted-framed submanifolds, is made explicit in:

A general equivariant Pontrjagin theorem – relating equivariant Cohomotopy to normal equivariant framed submanifolds – remains elusive, but on free G-manifolds it is again straightforward (and reduces to the twisted Pontrjagin theorem on the quotient space), made explicit in:

  • James Cruickshank, Thm. 5.0.6, Cor. 6.0.13 in: Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)
In negative codimension

In negative codimension, the Cohomotopy charge map from the Pontrjagin theorem gives the May-Segal theorem, now identifying Cohomotopy cocycle spaces with configuration spaces of points:

  • Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)

  • Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

    c Generalization of these constructions and results is due to

  • Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

  • Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)

Thom’s construction

Thom's theorem i.e. the unstable and oriented version of the Pontrjagin-Thom construction, identifying cobordism classes of normally oriented submanifolds with homotopy classes of maps to the universal special orthogonal Thom space MSO(n)M SO(n), is due to:

Textbook accounts:

Lashof’s construction

The joint generalization of Pontryagin 38a, 55 (framing structure) and Thom 54 (orientation structure) to any family of tangential structures (“(B,f)-structure”) is first made explicit in

and the general statement that has come to be known as Pontryagin-Thom isomorphism (identifying the stable cobordism classes of normally (B,f)-structure submanifolds with homotopy classes of maps to the Thom spectrum Mf) is Lashof 63, Theorem C.

Textbook accounts:

Lecture notes:

  • John Francis, Topology of manifolds course notes (2010) (web), Lecture 3: Thom’s theorem (pdf), Lecture 4 Transversality (notes by I. Bobkova) (pdf)

  • Cary Malkiewich, Section 3 of: Unoriented cobordism and MOM O, 2011 (pdf)

  • Tom Weston, Part I of An introduction to cobordism theory (pdf)

See also:

Relation to divisors

Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:

Last revised on March 3, 2021 at 03:57:33. See the history of this page for a list of all contributions to it.