cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
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:
global equivariant bordism theory:
algebraic:
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
$MU$ is the universal Thom spectrum for complex vector bundles. It is the spectrum representing complex cobordism cohomology theory. It is the complex analog of MO.
MR cohomology theory, or real cobordism, (Landweber 68, Landweber 69) is the $\mathbb{Z}_2$-equivariant cohomology theory version of $MU$ complex cobordism cohomology theory.
The spectrum denoted $M U$ is, as a sequential spectrum, in degree $2 n$ given by the Thom space of the underlying real vector bundle of the complex universal vector bundle: the vector bundle that is associated by the defining representation of the unitary group $U(n)$ on $\mathbb{C}^n$ to the $U(n)$-universal principal bundle:
A priori this yields a sequential S2-spectrum, which is then turned into a sequential $S^1$-spectrum by taking the component spaces in odd degree to be the smash product of the circle $S^1$ with those in even degree.
This represents a complex oriented cohomology theory and indeed the universal one among these, see at universal complex orientation on MU.
The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:
The cohomology ring $M P({*})$ is the Lazard ring which is the universal coefficient ring for formal group laws, see at Milnor-Quillen theorem on MU .
The periodic version is sometimes written $PMU$.
The graded ring given by evaluating complex cobordism theory on the point is both the complex cobordism ring as well as the Lazard ring classifying formal group laws.
Evaluation of $MU$ on the point yields the complex cobordism ring, whose underlying group is
where the generator $x_i$ is in degree $2 i$.
This is due to (Milnor 60, Novikov 60, Novikov 62). A review is in (Ravenel theorem 1.2.18, Ravenel, ch. 3, theorem 3.1.5).
The formal group law associated with $MU$ as with any complex oriented cohomology theory is classified by a ring homomorphism $L \longrightarrow \pi_\bullet(MU)$ out of the Lazard ring.
This canonical homomorphism is an isomorphism
between the Lazard ring and the $MU$-cohomology ring, hence by theorem with the complex cobordism ring.
This is Quillen's theorem on MU. (e.g Lurie 10, lect. 7, theorem 1)
There is a canonical complex orientation on $MU$ obtained from the map
For $E$ a homotopy-commutative ring spectrum there is a bijection between complex orientation of $E$ and ring spectrum maps of the form
(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)
See also at complex orientation and MU.
For $X$ a manifold or a topological space, the $MU$-homology group $MU_\ast(X)$ of its underlying homotopy type is the group of equivalence classes of maps $\Sigma \to X$ from manifolds $\Sigma$ with complex structure on the stable normal bundle, modulo suitable complex cobordisms.
See Ravenel chapter 1, section 2.
For more information, see the article bordism homology theory, which treats the oriented case; the case of (stable almost) complex structure is similar.
$MU$-cohomology groups of a manifold $M$ can be expressed in terms of bordisms given by proper complex-oriented maps into $M$.
For more information, see the article cobordism cohomology theory.
Moreover, the dual $MU$-Steenrod algebra $MU_\bullet(MU)$ forms a commutative Hopf algebroid over the Lazard ring. This is the content of the Landweber-Novikov theorem.
Snaith's theorem asserts that the periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized at the Bott element $\beta$:
The p-localization of $MU$ decomposes into the Brown-Peterson spectra.
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:
global equivariant bordism theory:
algebraic:
Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)
Pierre Conner, Edwin Floyd, Section 12 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Robert Stong, Chapter VII of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)
Pierre Conner, Larry Smith, On the complex bordism of finite complexes, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (numdam:PMIHES_1969__37__117_0)
Larry Smith, On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres, American Journal of Mathematics Vol. 92, No. 4 (Oct., 1970), pp. 793-856 (doi:10.2307/2373397)
Peter Landweber, On the complex bordism and cobordism of infinite complexes, Bull. Amer. Math. Soc. Volume 76, Number 3 (1970) (Euclid)
Frank Adams, Stable homotopy theory and generalized homology, Chicago lectures in mathematics, 1974
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986
Stanley Kochman, Section 4.4 of: Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs, American Mathematical Society, 1996 (cds:2264210)
Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, section VIII of Rings, modules and algebras in stable homotopy theory 1997 (pdf)
Mike Hopkins, section 4 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)
Dai Tamaki, Akira Kono, Section 3.7 and Chapter 6 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes), Lecture 5 Complex bordism
(pdf)
Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes) Lecture 6 MU and complex orientations (pdf)
Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes), Lecture 7 The homology of MU (pdf)
Neil Strickland, Products on $MU$-modules (pdf)
Jesse McKeown, Complex Cobordism vs. Representing Formal Group Laws (arXiv:1605.09252)
For general discussion of equivariant complex oriented cohomology see at equivariant cohomology – References – Complex oriented cohomology
On the Chern-Dold character on complex cobordism:
Victor Buchstaber, The Chern–Dold character in cobordisms. I,
Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (mathnet:3530)
English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 (doi:10.1070/SM1970v012n04ABEH000939)
Victor Buchstaber, A. P. Veselov, Chern-Dold character in complex cobordisms and abelian varieties (arXiv:2007.05782)
A relation to 2d CFT over Spec(Z) was suggested in
Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:
Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467-493 (doi:10.1090/S0894-0347-97-00232-4)
Last revised on October 19, 2022 at 20:21:45. See the history of this page for a list of all contributions to it.