In the field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the signature is an integer
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
which is defined for an oriented
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 n ...
''M'' of dimension
divisible by four.
This invariant of a manifold has been studied in detail, starting with
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its interse ...
for 4-manifolds, and
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combi ...
.
Definition
Given a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
and
oriented manifold ''M'' of dimension 4''k'', the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
gives rise to a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
''Q'' on the 'middle' real
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
:
.
The basic identity for the cup product
:
shows that with ''p'' = ''q'' = 2''k'' the product is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. It takes values in
:
.
If we assume also that ''M'' is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
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 ...
identifies this with
:
which can be identified with
. Therefore the cup product, under these hypotheses, does give rise to a
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
on ''H''
2''k''(''M'',''R''); and therefore to a quadratic form ''Q''. The form ''Q'' is
non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...
due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
with ''4n''-dimensional Poincaré duality.
The signature of ''M'' is by definition the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of ''Q'', an ordered triple according to its definition. If ''M'' is not connected, its signature is defined to be the sum of the signatures of its connected components.
Other dimensions
If ''M'' has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in
L-theory
In mathematics, algebraic ''L''-theory is the ''K''-theory of quadratic forms; the term was coined by C. T. C. Wall,
with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as "Hermitian ''K''-theory",
is important in ...
: the signature can be interpreted as the 4''k''-dimensional (simply connected) symmetric L-group
or as the 4''k''-dimensional quadratic L-group
and these invariants do not always vanish for other dimensions. The
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphe ...
is a mod 2 (i.e., an element of
) for framed manifolds of dimension 4''k''+2 (the quadratic L-group
), while the
de Rham invariant In geometric topology, the de Rham invariant is a mod 2 invariant of a (4''k''+1)-dimensional manifold, that is, an element of \mathbf/2 – either 0 or 1. It can be thought of as the simply-connected ''symmetric'' L-group L^, and thus analogous t ...
is a mod 2 invariant of manifolds of dimension 4''k''+1 (the symmetric L-group
); the other dimensional L-groups vanish.
Kervaire invariant
When
is twice an odd integer (
singly even
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Gree ...
), the same construction gives rise to an
antisymmetric bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear ...
. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a
quadratic refinement
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''.
Mathematics ...
of the form, which occurs if one has a
framed manifold, then the resulting
ε-quadratic form
In mathematics, specifically the theory of quadratic forms, an ''ε''-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; , accordingly for symmetric or skew-symmetric. They are also called (-)^n-qua ...
s need not be equivalent, being distinguished by the
Arf invariant. The resulting invariant of a manifold is called the
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphe ...
.
Properties
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...
(1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its
Pontryagin numbers
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
.
For example, in four dimensions, it is given by
.
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
(1954) found an explicit expression for this linear combination as the
L genus of the manifold.
William Browder (1962) proved that a simply connected compact
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
with 4''n''-dimensional
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 ...
is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combi ...
.
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its interse ...
says that the signature of a 4-dimensional simply connected manifold with a
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
is divisible by 16.
See also
*
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combi ...
*
Genus of a multiplicative sequence
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the ...
*
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its interse ...
References
{{DEFAULTSORT:Signature (Topology)
Geometric topology
Quadratic forms