“Novikov ring”的意思、由来-开放百科全书

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Novikov ring

释义

Novikov numbers
Notes
References
External links

For a concept in quantum cohomology, see the linked article.

In mathematics, given an additive subgroup , the Novikov ring of is the subring of ^[1] consisting of formal sums

such that and . The notion was introduced by S. P. Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function.

The Novikov ring is a principal ideal domain. Let S be the subset of consisting of those with leading term 1. Since the elements of S are unit elements of , the localization of with respect to S is a subring of called the "rational part" of ; it is also a principal ideal domain.

Novikov numbers

Given a smooth function f on a smooth manifold M with nondegenerate critical points, the usual Morse theory constructs a free chain complex such that the (integral) rank of is the number of critical points of f of index p (called the Morse number). It computes the homology of M: (cf. Morse homology.)

In an analogy with this, one can define "Novikov numbers". Let X be a connected polyhedron with a base point. Each cohomology class may be viewed as a linear functional on the first homology group and, composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism . By the universal property, this map in turns gives a ring homomorphism , making a module over . Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a -module. Let be a local coefficient system corresponding to with module structure given by . The homology group is a finitely generated module over which is, by the structure theorem, a direct sum of the free part and the torsion part. The rank of the free part is called the Novikov Betti number and is denoted by . The number of cyclic modules in the torsion part is denoted by . If , is trivial and is the usual Betti number of X.

The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.)

Notes

1. ^Here,

is the ring consisting of the formal sums

integers and t a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring

References

{{cite book | title=Topology of closed one-forms | volume=108 | series=Mathematical surveys and monographs | first=Michael | last=Farber | publisher=American Mathematical Society | year=2004 | isbn=0-8218-3531-9 | zbl=1052.58016 }}
S. P. Novikov, Multi-valued functions and functionals: An analogue of Morse theory. Soviet Mathematics - Doklady 24 (1981), 222–226.
S. P. Novikov: The Hamiltonian formalism and a multi-valued analogue of Morse theory. Russian Mathematical Surveys 35:5 (1982), 1–56.

External links

http://mathoverflow.net/questions/13203/different-definitions-of-novikov-ring

3 : Commutative algebra|Ring theory|Morse theory

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