In mathematics, given an additive subgroup
Γ
⊂
R
, the Novikov ring
Nov
(
Γ
)
of
Γ
is the subring of
Z
[
[
Γ
]
]
consisting of formal sums
∑
n
γ
i
t
γ
i
such that
γ
1
>
γ
2
>
⋯
and
γ
i
→
−
∞
. 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
Nov
(
Γ
)
is a principal ideal domain. Let S be the subset of
Z
[
Γ
]
consisting of those with leading term 1. Since the elements of S are unit elements of
Nov
(
Γ
)
, the localization
Nov
(
Γ
)
[
S
−
1
]
of
Nov
(
Γ
)
with respect to S is a subring of
Nov
(
Γ
)
called the "rational part" of
Nov
(
Γ
)
; it is also a principal ideal domain.
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
)
)
≈
H
∗
(
M
,
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
ξ
∈
H
1
(
X
,
R
)
may be viewed as a linear functional on the first homology group
H
1
(
X
,
R
)
and, composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism
ξ
:
π
=
π
1
(
X
)
→
R
. By the universal property, this map in turns gives a ring homomorphism
ϕ
ξ
:
Z
[
π
]
→
Nov
=
Nov
(
R
)
, making
Nov
a module over
Z
[
π
]
. Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a
Z
[
π
]
-module. Let
L
ξ
be a local coefficient system corresponding to
Nov
with module structure given by
ϕ
ξ
. The homology group
H
p
(
X
,
L
ξ
)
is a finitely generated module over
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
(
ξ
)
. The number of cyclic modules in the torsion part is denoted by
q
p
(
ξ
)
. If
ξ
=
0
,
L
ξ
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.)
