Novikov ring explained
In mathematics, given an additive subgroup
, the
Novikov ring \operatorname{Nov}(\Gamma)
of
is the subring of
[1] consisting of
formal sums
such that
and
. 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
\operatorname{Nov}(\Gamma)
is a
principal ideal domain. Let
S be the subset of
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
with nondegenerate critical points, the usual
Morse theory constructs a free
chain complex
such that the (integral) rank of
is the number of critical points of
f of index
p (called the Morse number). It computes the (integral)
homology of
(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
may be viewed as a linear functional on the first homology group
; when composed with the
Hurewicz homomorphism, it can be viewed as a group homomorphism
\xi\colon\pi=\pi1(X)\to\R
. By the universal property, this map in turns gives a ring homomorphism,
\phi\xi\colon\Z[\pi]\to\operatorname{Nov}=\operatorname{Nov}(\R)
, making
a module over
. Since
X is a
connected polyhedron, a
local coefficient system over it corresponds one-to-one to a
-module. Let
be a local coefficient system corresponding to
with module structure given by
. The homology group
is a finitely generated module over
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
. The number of cyclic modules in the torsion part is denoted by
. If
,
is trivial and
is the usual Betti number of
X.
The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.)
References
- Book: Farber, Michael . Topology of closed one-forms . 108 . Mathematical surveys and monographs . . 2004 . 0-8218-3531-9 . 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
Notes and References
- Here,
is the ring consisting of the formal sums
,
integers and t a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring
.
