cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
homotopy classes of maps to Thom space MO
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theoryM(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
What is called cohomology theory or Real cobordism (Landweber 68, Landweber 69) is the -equivariant cohomology theory version of complex cobordism .
There is an evident action of on formal group laws given by negation in the formal group (the inversion involution), and this lifts to an involutive automorphism of the spectrum MU. This induces an -equivariant spectrum , and real cobordism is the cohomology theory that it represents. This is directly analogous to how complex K-theory KU gives -equivariant KR-theory, both are examples of real-oriented cohomology theories.
A modern review in in (Kriz 01, section 2).
While is not the cobordism ring of real manifolds, still every real manifold does give a class in (Hu 99, Kriz 01, p. 13).
is naturally an E-∞ ring spectrum. (reviewed as Kriz 01, prop. 3.1)
In direct analogy with the situation for complex cobordism theory in complex oriented cohomology theory, is the univeral real oriented cohomology theory:
Equivalence classes of real orientations of a -equivariant E-∞ ring are in bijection to equivalence classes of E-∞ ring homomorphisms
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theoryM(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
The definition of Real cobordism cohomology goes back to:
Peter Landweber, Fixed point free conjugations on complex manifolds, Annals of Mathematics 86 (2) (1967) 491-502 [jstor:]
Peter Landweber, Conjugations on complex manifolds and equivariant homotopy of , Bulletin of the American Mathematical Society 74 (1968) 271-274 [Euclid]
and in the broader context of real-oriented cohomology theories:
Shôrô Araki, -Cohomology Theories, Japanese Journal of Mathematics 4 2 (1978) 363-416 [doi:10.4099/math1924.4.363]
Shôrô Araki, Forgetful spectral sequences, Osaka Journal of Mathematics 16 1 (1979) 173-199 [Euclid]
Shôrô Araki, Orientations in -cohomology theories, Japan Journal of Mathematics (N.S.) 5 2 (1979) 403-430 [doi:10.4099/math1924.5.403]
The Adams spectral sequence for Real cobordism:
Po Hu, The cobordism of Real manifolds and calculations with the Real Adams-Novikov spectral sequence, (1998) [hdl:2027.42/130996, pdf]
Po Hu, Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317-399 [doi:10.1016/S0040-9383(99)00065-8, pdf]
Further discussion:
Po Hu, The cobordism of Real manifolds, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 [dml:212395, pdf]
Po Hu, Igor Kriz, Some Remarks on Real and Algebraic Cobordism, K-Theory 22 (2001) 335–366 [pdf, doi:10.1023/A:1011196901303]
Po Hu, Igor Kriz, Topological Hermitian Cobordism, Journal of Homotopy and Related Structures, 11 (2016) 173–197 [arXiv:1110.5608, doi:10.1007/s40062-014-0100-9]
Last revised on October 22, 2023 at 17:34:11. See the history of this page for a list of all contributions to it.