nLab
cobordism cohomology theory

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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 due to Quillen 71, 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 theoryMB\,M B (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:

Pontryagin-Thom construction

Pontryagin’s construction

The unstable and framed version of the Pontrjagin-Thom construction, namely the original Pontryagin construction, identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge in unstable Cohomotopy sets, is due to:

  • Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

  • Lev Pontrjagin, Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

Textbook accounts:

Also:

Discussion of the early history:

Lecture notes:

  • Mladen Bestvina (notes by Adam Keenan), Chapter 16 in Differentiable Topology and Geometry, 2002 (pdf, pdf)

Thom’s construction

The stable and structured version of Pontryagin’s construction – the Pontryagin-Thom collapse – relating abstract (instead of embedded) cobordism classes to homotopy classes of maps to Thom spectra:

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 January 11, 2021 at 06:05:38. See the history of this page for a list of all contributions to it.