Thom space

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.

Construction of the Thom space

One way to construct this space is as follows. Let

p

:

E

→

B

{\displaystyle p\colon E\to B}

be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber

E

b

{\displaystyle E_{b}}

is an n-dimensional real vector space. We can form an n-sphere bundle

Sph

⁡

(

E

)

→

B

{\displaystyle \operatorname {Sph} (E)\to B}

by taking the one-point compactification of each fiber and gluing them together to get the total space. Finally, from the total space

Sph

⁡

(

E

)

{\displaystyle \operatorname {Sph} (E)}

we obtain the Thom space

T

(

E

)

{\displaystyle T(E)}

as the quotient of

Sph

⁡

(

E

)

{\displaystyle \operatorname {Sph} (E)}

by B; that is, by identifying all the new points to a single point

∞

{\displaystyle \infty }

, which we take as the basepoint of

T

(

E

)

{\displaystyle T(E)}

. If B is compact, then

T

(

E

)

{\displaystyle T(E)}

is the one-point compactification of E.

For example, if E is the trivial bundle

B

×

R

n

{\displaystyle B\times \mathbb {R} ^{n}}

, then

Sph

⁡

(

E

)

{\displaystyle \operatorname {Sph} (E)}

is

B

×

S

n

{\displaystyle B\times S^{n}}

and, writing

B

+

{\displaystyle B_{+}}

for B with a disjoint basepoint,

T

(

E

)

{\displaystyle T(E)}

is the smash product of

B

+

{\displaystyle B_{+}}

and

S

n

{\displaystyle S^{n}}

; that is, the n-th reduced suspension of

B

+

{\displaystyle B_{+}}

.

Alternatively, since B is paracompact, E can be given a Euclidean metric and then

T

(

E

)

{\displaystyle T(E)}

can be defined as the quotient of the unit disk bundle of E by the unit

(

n

−

1

)

{\displaystyle (n-1)}

-sphere bundle of E.

The Thom isomorphism

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

Z

2

{\displaystyle \mathbb {Z} _{2}}

coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)

Let

p

:

E

→

B

{\displaystyle p:E\to B}

be a real vector bundle of rank n. Then there is an isomorphism called a Thom isomorphism

Φ

:

H

k

(

B

;

Z

2

)

→

H

~

k

+

n

(

T

(

E

)

;

Z

2

)

,

{\displaystyle \Phi :H^{k}(B;\mathbb {Z} _{2})\to {\widetilde {H}}^{k+n}(T(E);\mathbb {Z} _{2}),}

for all k greater than or equal to 0, where the right hand side is reduced cohomology.

This theorem was formulated and proved by René Thom in his famous 1952 thesis.