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Examples ofcobordism
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He also worked on complexcobordism, showing that its formal group law is essentially the universal one.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Generalizations such ascobordismalso have induced homomorphisms.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This is the composition law for morphisms in thecobordismcategory.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This is similar to thecobordismring of oriented manifolds, except that the manifolds also have a vector bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These maps have to satisfy various conditions similar to those satisfied by complexcobordism.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Over a field of characteristic 0, algebraiccobordismis the universal oriented cohomology theory for smooth varieties.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In low dimensions, the bordism question is trivial, but the category ofcobordismis still interesting.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The resultingcobordismgroups are then defined analogously to the unoriented case.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It is a 1-dimensionalcobordismbetween the 0-dimensional manifolds 0, 1.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Contrarily tocobordismclasses of maps, the normal invariants are a "cohomology theory".
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This example is from Wikipedia and may be reused under a CC BY-SA license.Othercobordismtheories do not reduce to ordinary homology in this way, notably framedcobordism, orientedcobordismand complexcobordism.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecobordismclasses of manifolds form a ring that is usually the coefficient ring of some generalized cohomology theory.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Some of these cohomology theories, in particular complexcobordism, turned out to be some of the most powerful cohomology theories known.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The conditions on can be rephrased as saying that is a ring homomorphism from thecobordismring of manifolds (with given structure) to another ring.
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This example is from Wikipedia and may be reused under a CC BY-SA license.There exists a homotopy theoretic analogue of the s-cobordismtheorem.
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This example is from Wikipedia and may be reused under a CC BY-SA license.He also worked in the theory of characteristic classes, homotopy theory andcobordismtheory.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.