![Cobordism](https://upload.wikimedia.org/wikipedia/commons/4/49/Cobordism.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a genus of a multiplicative sequence is a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preservi ...
from the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of smooth
compact manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example is ...
s up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
) to another ring, usually the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, having the property that they are constructed from a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
Definition
A genus
assigns a number
to each manifold ''X'' such that
#
(where
is the disjoint union);
#
;
#
if ''X'' is the boundary of a manifold with boundary.
The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see
list of cobordism theories for many more examples). The value
is in some ring, often the ring of rational numbers, though it can be other rings such as
or the ring of modular forms.
The conditions on
can be rephrased as saying that
is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.
Example: If
is 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 the oriented manifold ''X'', then
is a genus from oriented manifolds to the ring of integers.
The genus associated to a formal power series
A sequence of polynomials
in variables
is called multiplicative if
:
implies that
:
If
is a
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
in ''z'' with constant term 1, we can define a multiplicative sequence
:
by
:
,
where
is the ''k''th
elementary symmetric function of the indeterminates
. (The variables
will often in practice be
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Definition
Given a real vector bundle ...
es.)
The genus
of
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 ...
,
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 ...
,
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
,
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
manifolds corresponding to ''Q'' is given by
:
where the
are the
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Definition
Given a real vector bundle ...
es of ''X''. The power series ''Q'' is called the characteristic power series of the genus
. A theorem of
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 ...
, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4''k'' for positive integers ''k'', implies that this gives a bijection between formal power series ''Q'' with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.
L genus
The L genus is the genus of the formal power series
:
where the numbers
are the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. The first few values are:
:
(for further ''L''-polynomials see or ). Now let ''M'' be a closed smooth oriented manifold of dimension 4''n'' with
Pontrjagin classes
.
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 ...
showed that the ''L'' genus of ''M'' in dimension 4''n'' evaluated on the
fundamental class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamen ...
of
, denoted