Manifolds and cobordisms
Special and general types
For a manifold, the group is the group of equivalence classes of maps from manifolds with complex structure on the stable normal bundle, modulo suitable complex cobordisms.
e.g (Ravenel chapter 1, section 2)
As represented by a spectrum
Cobordism cohomology theory, denoted for oriented cobordism cohomology , for complex cobordism cohomology etc, is the generalized (Eilenberg-Steenrod) cohomology theory represented by the universal Thom spectrum.
This spectrum, also denoted is the spectrum is in degree given by the Thom space of the vector bundle that is associated by the defining representation of the unitary group on to the universtal -principal bundle:
The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:
There is a canonical orientation? on this obtained from the map
this is the universal even periodic cohomology theory with orientation
The cohomology ring is the Lazard ring which is the universal coefficient ring for formal group laws.
The periodic version is sometimes written .
Differential cohomology refinement
The refinement of cobordism cohomology theory to differential cohomology is differential cobordism cohomology.
On the point: Cobordism and Lazard ring
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 on the point yields the complex cobordism ring, whose underlying group is
where the generator is in degree .
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 as with any complex oriented cohomology theory is classified by a ring homomorphism out of the Lazard ring.
This is Quillen's theorem on MU. (e.g Lurie 10, lect. 7, theorem 1)
On itself: Hopf algebroid structure on dual Steenrod algebra
Moreover, the dual -Steenrod algebra forms a commutative Hopf algebroid over the Lazard ring. This is the content of the Landweber-Novikov theorem.
Universal complex orientation
For an E-infinity ring there is a bijection between complex orientation of and E-infinity ring maps of the form
(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)
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 :
-Localization and Brown-Peterson spectrum
The p-localization of decomoses into the Brown-Peterson spectra.
Original articles include
- John Milnor, On the cobordism ring and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
- Sergei Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).
- Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442.
Textbooks accounts include
with an emphasis on the use of in the Adams-Novikov spectral sequence, and
- Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)
Lecture notes include
For further context see also the discussion at
Higher algebra over