Novikov ring

In mathematics, given an additive subgroup ${\displaystyle \Gamma \subset \mathbb {R} }$, the Novikov ring ${\displaystyle \operatorname {Nov} (\Gamma )}$ of ${\displaystyle \Gamma }$ is the subring of ${\displaystyle \mathbb {Z} [\![\Gamma ]\!]}$[1] consisting of formal sums ${\displaystyle \sum n_{\gamma _{i}}t^{\gamma _{i}}}$ such that ${\displaystyle \gamma _{1}>\gamma _{2}>\cdots }$ and ${\displaystyle \gamma _{i}\to -\infty }$. 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 notion is used in quantum cohomology, among the others.

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

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 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

• Different definitions of Novikov ring?