In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes of manifolds, the multiplication is given by the Cartesian...

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally...

cobordism" and "cobordism with G-structure", respectively. Under favourable technical conditions these form a graded ring called the cobordism ring Ω...

derived. Cobordism studies manifolds, where a manifold is regarded as "trivial" if it is the boundary of another compact manifold. The cobordism classes...

transversality prevents from computing cobordism rings of, say, topological manifolds from Thom spectra. Cobordism Cohomology operation Steenrod problem...

In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine...

For example, the Lazard ring is the ring of cobordism classes of complex manifolds. A graded-commutative ring with respect to a grading by Z/2 (as opposed...

bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are...

{\displaystyle \Phi } is a ring homomorphism R → C {\displaystyle R\to \mathbb {C} } , where R {\displaystyle R} is Thom's oriented cobordism ring. The genus Φ {\displaystyle...

structure. They have application in the cobordism ring in algebraic topology. Let Kn be polynomials over a ring A in indeterminates p1, ... weighted so...

boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds...

singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory. Highly structured ring spectrum Adams, J. Frank (1974), Stable...

construction, Gromov and Lawson observed that the h-cobordism theorem and analysis of the cobordism ring can be directly applied. They proved that, in dimension...

coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so...

cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring Ω U ∗ ( { p t } ) {\displaystyle \Omega _{U}^{*}(\{pt\})}...

and involved cobordism theory and pseudodifferential operators. The idea of this first proof is roughly as follows. Consider the ring generated by pairs...

the genus of a multiplicative sequence. Since the rational oriented cobordism ring Ω ∗ SO ⊗ Q {\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q}...

theorem, introduced the notions of oriented and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain...

manifold. For context see timeline of manifolds. Jean Dieudonné wrote that cobordism returns to the attempt in 1895 to define homology theory using only (smooth)...

homeomorphism, respectively), the s-cobordism theorem states that if the manifolds are not simply-connected, an h-cobordism is a cylinder if and only if the...

Due to its special properties it is of interest in algebraic topology, cobordism theory and spin geometry. The manifold was first studied and named after...

are not assumed to be simply connected, then an h-cobordism need not be a cylinder. The s-cobordism theorem, due independently to Mazur, Stallings, and...

that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring, explaining the unusual grading. A formal...

Brown–Peterson cohomology BP is a summand of MU(p), which is complex cobordism MU localized at a prime p. In fact MU(p) is a wedge product of suspensions...

Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and C. T. C. Wall used the spectral sequence...

oriented cobordism M S O ∗ ( X ) , {\displaystyle MSO^{*}(X),} complex cobordism M U ∗ ( X ) , {\displaystyle MU^{*}(X),} and so on. Complex cobordism has...

commutative rings. Another canonical example of spectra come from the Thom spectra representing various cobordism theories. This includes real cobordism M O {\displaystyle...

who used them to give explicit generators for René Thom's unoriented cobordism ring. Note that P ( m , 0 ) = R P m {\displaystyle P(m,0)=\mathbb {RP} ^{m}}...

theorem Signature (topology) Handle decomposition Handlebody h-cobordism theorem s-cobordism theorem Manifold decomposition Hilbert-Smith conjecture Mapping...

cohomology of groups and algebraic K-theory. He also worked on complex cobordism, showing that its formal group law is essentially the universal one. In...