Signature (topology)

Last updated April 23, 2023

In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.

Definition

Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group

H^{2k}(M,\mathbf {R} )

.

The basic identity for the cup product

\alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})

shows that with p = q = 2k the product is symmetric. It takes values in

H^{4k}(M,\mathbf {R} )

.

If we assume also that M is compact, Poincaré duality identifies this with

H^{0}(M,\mathbf {R} )

which can be identified with $\mathbf {R}$ . Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H^2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself.^[1] More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.

The signature $\sigma (M)$ of M is by definition the signature of Q, that is, $\sigma (M)=n_{+}-n_{-}$ where any diagonal matrix defining Q has $n_{+}$ positive entries and $n_{-}$ negative entries.^[2] 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: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group $L^{4k},$ or as the 4k-dimensional quadratic L-group $L_{4k},$ and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of $\mathbf {Z} /2$ ) for framed manifolds of dimension 4k+2 (the quadratic L-group $L_{4k+2}$ ), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group $L^{4k+1}$ ); the other dimensional L-groups vanish.

Kervaire invariant

When $d=4k+2=2(2k+1)$ is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. 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 of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.

Properties

Compact oriented manifolds M and N satisfy $\sigma (M\sqcup N)=\sigma (M)+\sigma (N)$ by definition, and satisfy $\sigma (M\times N)=\sigma (M)\sigma (N)$ by a Künneth formula.

If M is an oriented boundary, then $\sigma (M)=0$ .

René Thom (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.^[3] For example, in four dimensions, it is given by ${\frac {p_{1}}{3}}$ . Friedrich Hirzebruch (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 with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.

Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.

Related Research Articles

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As $K$ -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer $. That is,$

<span class="mw-page-title-main">Holonomy</span> Concept in differential geometry

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.

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 surgery theory.

In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.

<span class="mw-page-title-main">Symmetric space</span> A (pseudo-)Riemannian manifold whose geodesics are reversible.

In mathematics, a symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

<span class="mw-page-title-main">Genus of a multiplicative sequence</span>

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 to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension $, could be described as removing an imbedded sphere of dimension p from M . Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.$

In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure, then the signature of its intersection form, a quadratic form on the second cohomology group $, is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.$

In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.

In differential topology, an area of mathematics, the Hirzebruch signature theorem is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem.

In mathematics, the Kervaire invariant is an invariant of a framed $-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.$

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; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called $-quadratic forms, particularly in the context of surgery theory.$

In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of $- either 0 or 1. It can be thought of as the simply-connected symmetric L-group and thus analogous to the other invariants from L-theory: the signature, a 4 k -dimensional invariant, and the Kervaire invariant, a (4 k +2)-dimensional quadratic invariant$

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

This is a glossary of properties and concepts in algebraic topology in mathematics.

References

↑ Hatcher, Allen (2003). Algebraic topology (PDF) (Repr. ed.). Cambridge: Cambridge Univ. Pr. p. 250. ISBN 978-0521795401 . Retrieved 8 January 2017.
↑ Milnor, John; Stasheff, James (1962). Characteristic classes. Annals of Mathematics Studies 246. p. 224. CiteSeerX 10.1.1.448.869 . ISBN 978-0691081229.
↑ Thom, René. "Quelques proprietes globales des varietes differentiables" (PDF) (in French). Comm. Math. Helvetici 28 (1954), S. 17–86. Retrieved 26 October 2019.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Hatcher, Allen (2003). Algebraic topology (PDF) (Repr. ed.). Cambridge: Cambridge Univ. Pr. p. 250. ISBN 978-0521795401 . Retrieved 8 January 2017.

[2] Milnor, John; Stasheff, James (1962). Characteristic classes. Annals of Mathematics Studies 246. p. 224. CiteSeerX 10.1.1.448.869 . ISBN 978-0691081229.

[3] Thom, René. "Quelques proprietes globales des varietes differentiables" (PDF) (in French). Comm. Math. Helvetici 28 (1954), S. 17–86. Retrieved 26 October 2019.

[1]

[2]

[3]