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 ...
, the Thom space, Thom complex, or Pontryagin–Thom construction (named after
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 ...
and
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
) of
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
associated to 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 po ...
One way to construct this space is as follows. Let
:
be a rank ''n'' real
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 po ...
over the paracompact space ''B''. Then for each point ''b'' in ''B'', the
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
is an -dimensional real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let be the unit ball bundle with respect to our orthogonal structure, and let be the unit sphere bundle, then the Thom space is the quotient of topological spaces. is a pointed space with the image of in the quotient as basepoint. If ''B'' is compact, then is the one-point compactification of ''E''.
For example, if ''E'' is the trivial bundle , then and . Writing for ''B'' with a disjoint basepoint, is the
smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the quotient of the product space ''X'' × ''Y'' under the ide ...
of and ; that is, the ''n''-th reduced suspension of .
The Thom isomorphism
The significance of this construction begins with the following result, which belongs to the subject of
cohomology
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 ...
of
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s. (We have stated the result in terms of coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
Let be a real vector bundle of rank ''n''. Then there is an isomorphism, now called a Thom isomorphism
:
for all ''k'' greater than or equal to 0, where the
right hand side
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.reduced cohomology.
This theorem was formulated and proved by
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 ...
in his famous 1952 thesis.
We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on ''B'' of rank ''k'' is isomorphic to the ''k''th suspension of , ''B'' with a disjoint point added (cf. #Construction of the Thom space.) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:
In concise terms, the last part of the theorem says that ''u'' freely generates as a right -module. The class ''u'' is usually called the Thom class of ''E''. Since the pullback is a ring isomorphism, is given by the equation:
:
In particular, the Thom isomorphism sends the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
element of to ''u''. Note: for this formula to make sense, ''u'' is treated as an element of (we drop the ring )
:
Significance of Thom's work
In his 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the
Steenrod operation In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, ...
s were all related. He used these ideas to prove in the 1954 paper ''Quelques propriétés globales des variétés differentiables'' that the
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 ...
groups could be computed as the
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of certain Thom spaces ''MG''(''n''). The proof depends on and is intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei Novikov (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as surgery theory. In addition, the spaces ''MG(n)'' fit together to form spectra ''MG'' now known as Thom spectra, and the cobordism groups are in fact
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
. Thom's construction thus also unifies
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
and stable homotopy theory, and is in particular integral to our knowledge of the
stable homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...
.
If the Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel–Whitney classes. Recall that the Steenrod operations (mod 2) are natural transformations
:
defined for all nonnegative integers ''m''. If , then coincides with the cup square. We can define the ''i''th Stiefel–Whitney class of the vector bundle by:
:
Consequences for differentiable manifolds
If we take the bundle in the above to be the tangent bundle of a smooth manifold, the conclusion of the above is called the
Wu formula
Wu may refer to:
States and regions on modern China's territory
*Wu (state) (; och, *, italic=yes, links=no), a kingdom during the Spring and Autumn Period 771–476 BCE
** Suzhou or Wu (), its eponymous capital
** Wu County (), a former county i ...
, and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational
Pontryagin classes 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 bundl ...
There are two ways to think about bordism: one as considering two -manifolds are cobordant if there is an -manifold with boundary such that
:
Another technique to encode this kind of information is to take an embedding and considering the normal bundle
:
The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class