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. Vol. 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

[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 $\mathbb {Z} [\Gamma ]$ .

[1]