In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a
topological invariant
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s introduced by
Kurt Reidemeister
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...
for
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s and generalized to higher
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
s by and .
Analytic torsion (or Ray–Singer torsion) is an invariant of
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
s defined by as an analytic analogue of Reidemeister torsion. and proved Ray and Singer's conjecture that
Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
Reidemeister torsion was the first invariant in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
that could distinguish between closed manifolds which are
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
but not
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
, and can thus be seen as the birth of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated ...
as a distinct field. It can be used to classify
lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualized ...
s.
Reidemeister torsion is closely related to
Whitehead torsion In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \op ...
; see . It has also given some important motivation to
arithmetic topology; see . For more recent work on torsion see the books and .
Definition of analytic torsion
If ''M'' is a Riemannian manifold and ''E'' a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
over ''M'', then there is a
Laplacian operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
acting on the ''k''-forms with values in ''E''. If the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s on ''k''-forms are λ
''j'' then the zeta function ζ
''k'' is defined to be
:
for ''s'' large, and this is extended to all complex ''s'' by
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
.
The zeta regularized determinant of the Laplacian acting on ''k''-forms is
:
which is formally the product of the positive eigenvalues of the laplacian acting on ''k''-forms.
The analytic torsion ''T''(''M'',''E'') is defined to be
:
Definition of Reidemeister torsion
Let
be a finite connected
CW-complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
with
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
and
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
, and let
be an orthogonal finite-dimensional
-representation. Suppose that
:
for all n. If we fix a cellular basis for
and an orthogonal
-basis for
, then
is a contractible finite based free
-chain complex. Let
be any chain contraction of D
*, i.e.
for all
. We obtain an isomorphism
with
,
. We define the Reidemeister torsion
:
where A is the matrix of
with respect to the given bases. The Reidemeister torsion
is independent of the choice of the cellular basis for
, the orthogonal basis for
and the chain contraction
.
Let
be a compact smooth manifold, and let
be a unimodular representation.
has a smooth triangulation. For any choice of a volume
, we get an invariant
. Then we call the positive real number
the Reidemeister torsion of the manifold
with respect to
and
.
A short history of Reidemeister torsion
Reidemeister torsion was first used to combinatorially classify 3-dimensional
lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualized ...
s in by Reidemeister, and in higher-dimensional spaces by Franz. The classification includes examples of
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
3-dimensional manifolds which are not
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
— at the time (1935) the classification was only up to
PL homeomorphism
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear ...
, but later showed that this was in fact a classification up to
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
.
J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes. This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant.
Whitehead torsion In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \op ...
provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to the concept of "simple homotopy type", see
In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemeister torsion of its knot complement in
. For each ''q'' the
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
induces
:
and then we obtain
:
The representation of the fundamental group of knot complement plays a central role in them. It gives the relation between knot theory and torsion invariants.
Cheeger–Müller theorem
Let
be an orientable compact Riemann manifold of dimension n and
a representation of the fundamental group of
on a real vector space of dimension N. Then we can define the de Rham complex
:
and the formal adjoint
and
due to the flatness of
. As usual, we also obtain the Hodge Laplacian on p-forms
:
Assuming that
, the Laplacian is then a symmetric positive semi-positive elliptic operator with pure point spectrum
:
As before, we can therefore define a zeta function associated with the Laplacian
on
by
:
where
is the projection of
onto the kernel space
of the Laplacian
. It was moreover shown by that
extends to a meromorphic function of
which is holomorphic at
.
As in the case of an orthogonal representation, we define the analytic torsion
by
:
In 1971 D.B. Ray and I.M. Singer conjectured that
for any unitary representation
. This Ray–Singer conjecture was eventually proved, independently, by and . Both approaches focus on the logarithm of torsions and their traces. This is easier for odd-dimensional manifolds than in the even-dimensional case, which involves additional technical difficulties. This Cheeger–Müller theorem (that the two notions of torsion are equivalent), along with
Atiyah–Patodi–Singer theorem, later provided the basis for
Chern–Simons perturbation theory.
A proof of the Cheeger-Müller theorem for arbitrary representations was later given by J. M. Bismut and Weiping Zhang. Their proof uses the
Witten deformation.
References
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* Online book
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*{{Citation , last1=Seeley , first1=R. T. , editor1-last=Calderón , editor1-first=Alberto P. , title=Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) , publisher=Amer. Math. Soc. , location=Providence, R.I. , series=Proceedings of Symposia in Pure Mathematics , isbn=978-0-8218-1410-9 , mr=0237943 , year=1967 , volume=10 , chapter=Complex powers of an elliptic operator , pages=288–307
Differential geometry
3-manifolds
Surgery theory