Novikov ring

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

↑ 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

Farber, Michael (2004). Topology of closed one-forms. Mathematical surveys and monographs. 108. American Mathematical Society. ISBN 0-8218-3531-9. Zbl 1052.58016.
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

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

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