In mathematics, given an additive subgroup Γ ⊂ R {\displaystyle \Gamma \subset \mathbb {R} } , the Novikov ring Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} of Γ {\displaystyle \Gamma } is the subring of Z [ [ Γ ] ] {\displaystyle \mathbb {Z} [\![\Gamma ]\!]} consisting of formal sums ∑ n γ i t γ i {\displaystyle \sum n_{\gamma _{i}}t^{\gamma _{i}}} such that γ 1 > γ 2 > ⋯ {\displaystyle \gamma _{1}>\gamma _{2}>\cdots } and γ i → − ∞ {\displaystyle \gamma _{i}\to -\infty } . The notion was introduced by Sergei 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 Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} is a principal ideal domain. Let S be the subset of Z [ Γ ] {\displaystyle \mathbb {Z} [\Gamma ]} consisting of those with leading term 1. Since the elements of S are unit elements of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} , the localization Nov ( Γ ) [ S − 1 ] {\displaystyle \operatorname {Nov} (\Gamma )[S^{-1}]} of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} with respect to S is a subring of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} called the "rational part" of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} ; it is also a principal ideal domain.
Novikov numbers Given a smooth function f on a smooth manifold M {\displaystyle M} with nondegenerate critical points, the usual Morse theory constructs a free chain complex C ∗ ( f ) {\displaystyle C_{*}(f)} such that the (integral) rank of C p {\displaystyle C_{p}} is the number of critical points of f of index p (called the Morse number). It computes the (integral) homology of M {\displaystyle M} (cf. Morse homology):
H ∗ ( C ∗ ( f ) ) ≅ H ∗ ( M , Z ) {\displaystyle H^{*}(C_{*}(f))\cong H^{*}(M,\mathbb {Z} )}
In an analogy with this, one can define "Novikov numbers". Let X be a connected polyhedron with a base point. Each cohomology class ξ ∈ H 1 ( X , R ) {\displaystyle \xi \in H^{1}(X,\mathbb {R} )} may be viewed as a linear functional on the first homology group H 1 ( X , R ) {\displaystyle H_{1}(X,\mathbb {R} )} ; when composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism ξ : π = π 1 ( X ) → R {\displaystyle \xi \colon \pi =\pi _{1}(X)\to \mathbb {R} } . By the universal property, this map in turns gives a ring homomorphism,
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