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Novikov ring

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Mathematical construct

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} \!]} 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} } 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 )} 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,

ϕ ξ : Z [ π ] → Nov = Nov ⁡ ( R ) {\displaystyle \phi _{\xi }\colon \mathbb {Z} \to \operatorname {Nov} =\operatorname {Nov} (\mathbb {R} )} ,

making Nov {\displaystyle \operatorname {Nov} } a module over Z [ π ] {\displaystyle \mathbb {Z} } . Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a Z [ π ] {\displaystyle \mathbb {Z} } -module. Let L ξ {\displaystyle L_{\xi }} be a local coefficient system corresponding to Nov {\displaystyle \operatorname {Nov} } with module structure given by ϕ ξ {\displaystyle \phi _{\xi }} . The homology group H p ( X , L ξ ) {\displaystyle H_{p}(X,L_{\xi })} is a finitely generated module over Nov , {\displaystyle \operatorname {Nov} ,} which is, by the structure theorem, the direct sum of its free part and its torsion part. The rank of the free part is called the Novikov Betti number and is denoted by b p ( ξ ) {\displaystyle b_{p}(\xi )} . The number of cyclic modules in the torsion part is denoted by q p ( ξ ) {\displaystyle q_{p}(\xi )} . If ξ = 0 {\displaystyle \xi =0} , L ξ {\displaystyle L_{\xi }} is trivial and b p ( 0 ) {\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, Z [ [ Γ ] ] {\displaystyle \mathbb {Z} \!]} is the ring consisting of the formal sums ∑ γ ∈ Γ n γ t γ {\displaystyle \sum _{\gamma \in \Gamma }n_{\gamma }t^{\gamma }} , n γ {\displaystyle n_{\gamma }} integers and t a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring Z [ Γ ] {\displaystyle \mathbb {Z} } .

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 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.

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