In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
One way to construct this space is as follows. Let
p\colonE\toB
Eb
\operatorname{Sph}(E)\toB
\operatorname{Sph}(E)
T(E)
\operatorname{Sph}(E)
infty
T(E)
T(E)
For example, if E is the trivial bundle
B x \Rn
\operatorname{Sph}(E)
B x Sn
B+
T(E)
B+
Sn
B+
Alternatively, since B is paracompact, E can be given a Euclidean metric and then
T(E)
(n-1)
The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles. (We have stated the result in terms of
\Z2
Let
p:E\toB
\Phi:Hk(B;\Z2)\to\widetilde{H}k+n(T(E);\Z2),
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 kth suspension of
B+
In concise terms, the last part of the theorem says that u freely generates
H*(E,E\setminusB;Λ)
H*(E;Λ)
p*:H*(B;Λ)\toH*(E;Λ)
\Phi
\Phi(b)=p*(b)\smileu.
In particular, the Thom isomorphism sends the identity element of
H*(B)
Λ
\tilde{H}n(T(E))=Hn(\operatorname{Sph}(E),B)\simeqHn(E,E\setminusB).
The standard reference for the Thom isomorphism is the book by Bott and Tu.
In his 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, 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 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. 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 transformations
Sqi:Hm(-;\Z2)\toHm+i(-;\Z2),
defined for all nonnegative integers m. If
i=m
Sqi
wi(p)
p:E\toB
wi(p)=\Phi-1(Sqi(\Phi(1)))=\Phi-1(Sqi(u)).
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, 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.
There are two ways to think about bordism: one as considering two
n
M,M'
(n+1)
W
\partialW=M\coprodM'
Another technique to encode this kind of information is to take an embedding
M\hookrightarrow\RN
\nu:
N | |
\RN+n/M |
\toM
The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class
[M]
W
NW+n | |
\R |
x [0,1]
MO(n)
\pinMO\cong
O | |
\Omega | |
n |
requires a little more work.[3]
By definition, the Thom spectrum[4] is a sequence of Thom spaces
MO(n)=T(\gamman)
where we wrote
\gamman\toBO(n)
\pi*(MO)
\operatorname{Sph}(E)
(\operatorname{Sph}(E),\operatorname{Sph}(E)\setminusB,B)
\operatorname{Sph}(E)\setminusB
Hn(Sph(E),B)\simeqHn(\operatorname{Sph}(E),\operatorname{Sph}(E)\setminusB),
the latter of which is isomorphic to:
Hn(E,E\setminusB)
by excision.