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 is originally due to
Peter Landweber, Fixed point free conjugations on complex manifolds, Annals of Mathematics 86 (2) (1967) 491-502.
Peter Landweber, Conjugations on complex manifolds and equivariant homotopy of $MU$, Bulletin of the American Mathematical Society 74 (1968) 271-274
Computations in:
Po Hu, The cobordism of Real manifolds, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 (dml:212395, pdf)
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)
Last revised on November 21, 2020 at 13:37:17. See the history of this page for a list of all contributions to it.