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:
What is called $MR$ cohomology theory or Real cobordism (Landweber 68, Landweber 69) is the $\mathbb{Z}_2$-equivariant cohomology theory version of complex cobordism $MU$.
There is an evident action of $\mathbb{Z}_2$ on formal group laws given by negation in the formal group (the inversion involution), and this lifts to an involutive automorphism $MU \stackrel{\simeq}{\to} MU$ of the spectrum MU. This induces an $\mathbb{Z}_2$-equivariant spectrum $M\mathbb{R}$, and real cobordism is the cohomology theory that it represents. This is directly analogous to how complex K-theory KU gives $\mathbb{Z}_2$-equivariant KR-theory, both are examples of real-oriented cohomology theories.
A modern review in in (Kriz 01, section 2).
While $\pi_\bullet(M \mathbb{R})$ is not the cobordism ring of real manifolds, still every real manifold does give a class in $M \mathbb{R}$ (Hu 99, Kriz 01, p. 13).
$M \mathbb{R}$ 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, $M \mathbb{R}$ is the univeral real oriented cohomology theory:
Equivalence classes of real orientations of a $\mathbb{Z}/2\mathbb{Z}$-equivariant E-∞ ring $E$ 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 theory$\;$M(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 $MU$, Bulletin of the American Mathematical Society 74 (1968) 271-274 [Euclid]
and in the broader context of real-oriented cohomology theories:
Shôrô Araki, $\tau$-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 $\tau$-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]
Last revised on May 16, 2023 at 09:14:35. See the history of this page for a list of all contributions to it.