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# Bordism and cobordism theories

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description: Cobordism studies manifolds, where a manifold is regarded as "trivial" if it is the boundary of another compact manifold. The cobordism classes of manifolds form a ring that is usually the coefficient ...
 Cobordism studies manifolds, where a manifold is regarded as "trivial" if it is the boundary of another compact manifold. The cobordism classes of manifolds form a ring that is usually the coefficient ring of some generalized cohomology theory. There are many such theories, corresponding roughly to the different structures that one can put on a manifold.The functors of cobordism theories are often represented by Thom spaces of certain groups.Stable homotopy and cohomotopySpectrum: S (sphere spectrum).Coefficient ring: The coefficient groups πn(S) are the stable homotopy groups of spheres, which are notoriously hard to compute or understand for n > 0. (For n < 0 they vanish, and for n = 0 the group is Z.)Stable homotopy is closely related to cobordism of framed manifolds (manifolds with a trivialization of the normal bundle).Unoriented cobordismSpectrum: MO (Thom spectrum of orthogonal group)Coefficient ring: π*(MO) is the ring of cobordism classes of unoriented manifolds, and is a polynomial ring over the field with 2 elements on generators of degree i for every i not of the form 2n−1.Unoriented bordism is 2-torsion, since 2M is the boundary of M \times I.MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π*(MO)) ("homology with coefficients in π*(MO)") – MO is a product of Eilenberg–MacLane spectra. In other words the corresponding homology and cohomology theories are no more powerful than homology and cohomology with coefficients in Z/2Z. This was the first cobordism theory to be described completely.Complex cobordismMain article: Complex cobordismSpectrum: MU (Thom spectrum of unitary group)Coefficient ring: π*(MU) is the polynomial ring on generators of degree 2, 4, 6, 8, ... and is naturally isomorphic to Lazard's universal ring, and is the cobordism ring of stably almost complex manifolds.Oriented cobordism[icon]    This section requires expansion. (December 2009)Spectrum: MSO (Thom spectrum of special orthogonal group)Coefficient ring: The oriented cobordism class of a manifold is completely determined by its characteristic numbers: its Stiefel–Whitney numbers and Pontryagin numbers, but the overall coefficient ring, denoted \Omega_* = \Omega(*) = MSO(*) is quite complicated. Rationally, and at 2 (corresponding to Pontryagin and Stiefel–Whitney classes, respectively), MSO is a product of Eilenberg–MacLane spectra – MSO_{\mathbf Q} = H(\pi_*(MSO_{\mathbf Q})) and MSO[2] = H(\pi_*(MSO[2])) – but at odd primes it is not, and the structure is complicated to describe. The ring has been completely described integrally, due to work of Milnor, Averbuch, Rokhlin, and C. T. C. Wall.Special unitary cobordismSpectrum: MSU (Thom spectrum of special unitary group)Coefficient ring:Spin cobordism (and variants)Spectrum: MSpin (Thom spectrum of spin group)Coefficient ring: See (D. W. Anderson, E. H. Brown & F. P. Peterson 1967).Symplectic cobordismSpectrum: MSp (Thom spectrum of symplectic group)Coefficient ring:Clifford algebra cobordismPL cobordism and topological cobordismSpectrum: MPL, MSPL, MTop, MSTopCoefficient ring:The definition is similar to cobordism, except that one uses piecewise linear or topological instead of smooth manifolds, either oriented or unoriented. The coefficient rings are complicated.Brown–Peterson cohomologySpectrum: BPCoefficient ring: π*(BP) is a polynomial algebra over Z(p) on generators vn of dimension 2(pn − 1) for n ≥ 1.Brown–Peterson cohomology BP is a summand of MUp, which is complex cobordism MU localized at a prime p. In fact MU(p) is a sum of suspensions of BP.Morava K-theorySpectrum: K(n) (They also depend on a prime p.)Coefficient ring: Fp[vn, vn−1], where vn has degree 2(pn − 1).These theories have period 2(pn − 1). They are named after Jack Morava.Johnson–Wilson theorySpectrum E(n)Coefficient ring Z(2)[v1, ..., vn, 1/vn] where vi has degree 2(2i−1)String cobordismSpectrum:Coefficient ring:

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