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In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and differential topology 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 po ...
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 p ...
, over any
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
space.


Construction of the Thom space

One way to construct this space is as follows. Let :p: E \to B be a rank ''n''
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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 the
paracompact space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
''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 incorpora ...
E_b is an n-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 D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a
pointed space In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For example, if ''E'' is the trivial bundle B\times \R^n, then D(E) = B \times D^n and S(E) = B \times S^. Writing B_+ for ''B'' with a disjoint basepoint, T(E) 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 id ...
of B_+ and S^n; that is, the ''n''-th reduced
suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspende ...
of B_+.


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 bundles. (We have stated the result in terms of \Z_2
coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
to avoid complications arising from
orientability 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 i ...
; see also Orientation of a vector bundle#Thom space.) Let p: E\to B be a real vector bundle of rank ''n''. Then there is an isomorphism, now called a Thom isomorphism :\Phi : H^k(B; \Z_2) \to \widetilde^(T(E); \Z_2), for all ''k'' greater than or equal to 0, where the right hand side is
reduced cohomology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
. This theorem was formulated and proved by René Thom 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_+, ''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 H^*(E, E \setminus B; \Lambda) as a right H^*(E; \Lambda)-module. The class ''u'' is usually called the Thom class of ''E''. Since the pullback p^*: H^*(B; \Lambda) \to H^*(E; \Lambda) is a
ring isomorphism 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 ...
, \Phi is given by the equation: :\Phi(b) = p^*(b) \smile u. In particular, the Thom isomorphism sends the identity element of H^*(B) to ''u''. Note: for this formula to make sense, ''u'' is treated as an element of (we drop the ring \Lambda) :\tilde^n(T(E)) = H^n(\operatorname(E), B) \simeq H^n(E, E \setminus B).


Significance of Thom's work

In his 1952 paper, Thom showed that the Thom class, the
Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
es, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper ''Quelques propriétés globales des variétés differentiables'' that the cobordism 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, homo ...
of certain Thom spaces ''MG''(''n''). The proof depends on and is intimately related to the transversality properties of
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
—see
Thom transversality theorem In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It ...
. By reversing this construction,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
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 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 . Milnor called this technique ''surgery'', while And ...
. 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. Thom's construction thus also unifies differential topology and stable homotopy theory, and is in particular integral to our knowledge of the stable homotopy groups of spheres. 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 transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s :Sq^i : H^m(-; \Z_2) \to H^(-; \Z_2), defined for all nonnegative integers ''m''. If i=m, then Sq^i coincides with the cup square. We can define the ''i''th Stiefel–Whitney class w_i(p) of the vector bundle p: E\to B by: :w_i(p) = \Phi^(Sq^i(\Phi(1))) = \Phi^(Sq^i(u)).


Consequences for differentiable manifolds

If we take the bundle in the above to be the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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, due to Sergei Novikov.


Thom spectrum


Real cobordism

There are two ways to think about bordism: one as considering two n-manifolds M,M' are cobordant if there is an (n+1)-manifold with boundary W such that :\partial W = M \coprod M' Another technique to encode this kind of information is to take an embedding M \hookrightarrow \R^ and considering the normal bundle :\nu: N_ \to M The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class /math>. This can be shown by using a cobordism W and finding an embedding to some \R^\times ,1/math> which gives a homotopy class of maps to the Thom space MO(n) defined below. Showing the isomorphism of :\pi_nMO \cong \Omega^O_n requires a little more work.


Definition of Thom spectrum

By definition, the Thom spectrum is a sequence of Thom spaces :MO(n) = T(\gamma^n) where we wrote \gamma^n\to BO(n) for the universal vector bundle of rank ''n''. The sequence forms a
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
. A theorem of Thom says that \pi_*(MO) is the unoriented cobordism ring; the proof of this theorem relies crucially on Thom’s transversality theorem.http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf The lack of transversality prevents from computing cobordism rings of, say,
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
s from Thom spectra.


See also

* Cobordism *
Cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a coho ...
* Steenrod problem *
Hattori–Stong theorem In algebraic topology, the Hattori–Stong theorem, proved by and , gives an isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. T ...


Notes


References

* * A classic reference for differential topology, treating the link to
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 ...
and the
Euler class In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
of
Sphere bundle In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betw ...
s * * * * *


External links

*http://ncatlab.org/nlab/show/Thom+spectrum * {{springer, title=Thom space, id=p/t092680 * Akhil Mathew's blog posts: https://amathew.wordpress.com/tag/thom-space/ Algebraic topology Characteristic classes