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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 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 $f$ encoded in the representing Thom spectrum $M f$. 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 $X$, the cobordism cohomology group $\mathrm{M} \mathrm{U}^q(X) \;\coloneqq\; [\Sigma^\infty X_+, \Sigma^q MU]$ is equivalently the set of cobordism classes of proper complex-oriented maps $f \colon Z \to X$ of codimension $q$.

This uses the following definitions:

###### Definition

(complex-oriented maps)

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

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

$p \circ i \;\colon\; Z \longrightarrow E \longrightarrow X \,,$

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

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

###### Definition

(cobordism classes of maps)

Here two proper complex-oriented maps $f_i \colon Z_i \to X$ are called cobordant if there is a proper complex-oriented map $b\colon W\to X\times\mathbf{R}$ such that $X\times\{0\}$ and $X\times\{1\}$ are transversal to $b$ and pulling back $b$ to these submanifolds yields $f_0$ and $f_1$.

## Examples

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\,M B$ (B-bordism):

relative bordism theories:

algebraic:

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex 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 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:

Review:

Discussion of the early history:

#### 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 $M SO(n)$, is due to:

Textbook accounts:

#### Structured and Stable Pontryagin-Thom construction

By a Pontryagin-Thom isomorphism one typically means the generalization of the above to any other tangential structure and/or its stabilization involving 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 $M O$, 2011 (pdf)

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