Novikov ring

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

In mathematics, given an additive subgroup \Gamma \subset \mathbb{R}, the Novikov ring \operatorname{Nov}(\Gamma) of \Gamma is the subring of \mathbb{Z}[\![\Gamma]\!][1] consisting of formal sums \sum n_{\gamma_i} t^{\gamma_i} such that \gamma_1 > \gamma_2 > \cdots and \gamma_i \to -\infty. 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 \operatorname{Nov}(\Gamma) is a principal ideal domain. Let S be the subset of \mathbb{Z}[\Gamma] consisting of those with leading term 1. Since the elements of S are unit elements of \operatorname{Nov}(\Gamma), the localization \operatorname{Nov}(\Gamma)[S^{-1}] of \operatorname{Nov}(\Gamma) with respect to S is a subring of \operatorname{Nov}(\Gamma) called the "rational part" of \operatorname{Nov}(\Gamma); 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 C_*(f) such that the (integral) rank of C_p is the number of critical points of f of index p (called the Morse number). It computes the homology of M: H^*(C_*(f)) \approx H^*(M, \mathbf{Z}) (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 \xi \in H^1(X, \mathbb{R}) may be viewed as a linear functional on the first homology group H_1(X, \mathbb{R}) and, composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism \xi: \pi=\pi_1(X) \to \mathbb{R}. By the universal property, this map in turns gives a ring homomorphism \phi_\xi: \mathbb{Z}[\pi] \to \operatorname{Nov} = \operatorname{Nov}(\mathbb{R}), making \operatorname{Nov} a module over \mathbb{Z}[\pi]. Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a \mathbb{Z}[\pi]-module. Let L_\xi be a local coefficient system corresponding to \operatorname{Nov} with module structure given by \phi_\xi. The homology group H_p(X, L_\xi) is a finitely generated module over \operatorname{Nov}, 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 b_p(\xi). The number of cyclic modules in the torsion part is denoted by q_p(\xi). If \xi = 0, L_\xi is trivial and b_p(0) 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, \mathbb{Z}[\![\Gamma]\!] is the ring consisting of the formal sums \sum_{\gamma \in \Gamma} n_\gamma t^\gamma, n_\gamma integers and t a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring \mathbb{Z}[\Gamma].

References

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  • S. P. Novikov, Multi-valued functions and functionals: An analogue of Morse theory. Soviet Math. 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