Thom space and Thom isomorphism

Intuition [Beginner]

Given a vector bundle (a family of vector spaces parametrised by a base space), the Thom space is a one-point compactification of the total space. You take all the vectors in all the fibres and add a single point at infinity. If the base is compact, this is the same as collapsing the boundary sphere-bundle of the associated disk-bundle to a single point.

The amazing fact is the Thom isomorphism: if the bundle is oriented and has rank $n$, then the cohomology of the Thom space in dimension $k+n$ is isomorphic to the cohomology of the base in dimension $k$. The isomorphism is given by cup product with a special cohomology class called the Thom class.

Concretely, for the product rank-2 bundle over $S^1$, the Thom space is $S^1\times {\mathbb{R}}^2$ plus a point at infinity, which is $S^2\vee S^1$ (up to homotopy). The Thom isomorphism says $H^0(S^1)\cong H^2(\mathrm{Th})=\mathbb{Z}$ and $H^1(S^1)\cong H^3(\mathrm{Th})=0$, matching the cohomology of $S^2\vee S^1$.

Visual [Beginner]

A circle (base space $B$) with vertical lines (fibres) attached at each point, forming a cylinder. The disc bundle $D(\xi )$ is drawn as a solid tube around the circle. The sphere bundle $S(\xi )$ is the surface of the tube. The Thom space is obtained by collapsing this surface to a single point, creating a pinched sphere with a distinguished basepoint.

The Thom space captures the topology of the vector bundle in a single compact space whose cohomology encodes the twisting of the fibres.

Worked example [Beginner]

The tangent bundle of $S^2$. The tangent bundle $T{S}^2$ is a rank-2 oriented vector bundle over $S^2$. The Thom space $\mathrm{Th}(T{S}^2)$ is the one-point compactification of $T{S}^2$, which is homeomorphic to the quotient $D(T{S}^2)/S(T{S}^2)$.

The Thom isomorphism $\Phi :H^k(S^2)\to H^{k+2}(\mathrm{Th}(T{S}^2))$ gives:

• $\Phi :H^0(S^2)=\mathbb{Z}\to H^2(\mathrm{Th})=\mathbb{Z}$ (isomorphism)
• $\Phi :H^2(S^2)=\mathbb{Z}\to H^4(\mathrm{Th})=\mathbb{Z}$ (isomorphism)

The Thom class $u\in H^2(\mathrm{Th}(T{S}^2))$ restricts to the orientation class on each fibre. The Euler class $e(T{S}^2)={\Phi }^{-1}(u⌣u)=\chi (S^2)=2$ (the Euler characteristic), reflecting the fact that a tangent vector field on $S^2$ must vanish at least twice.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Thom space). Let $\xi =(E\stackrel{p}{\to }B)$ be a real vector bundle of rank $n$. The disk bundle $D(\xi )=\{v\in E:|v|\le 1\}$ and the sphere bundle $S(\xi )=\{v\in E:|v|=1\}$ (defined using a choice of Euclidean metric on the fibres). The Thom space of $\xi$ is the quotient:

$$\mathrm{Th}(\xi )=D(\xi )/S(\xi )$$

Equivalently, $\mathrm{Th}(\xi )$ is the one-point compactification of the total space $E$.

Definition (Thom class and Thom isomorphism). Let $\xi$ be an oriented real vector bundle of rank $n$ over $B$. A Thom class is a cohomology class $u\in H^n(\mathrm{Th}(\xi );\mathbb{Z})$ whose restriction to each fibre $H^n(\mathrm{Th}(\xi {|}_b);\mathbb{Z})\cong H^n(S^n;\mathbb{Z})=\mathbb{Z}$ is the preferred generator given by the orientation. The Thom isomorphism is the map:

$$\Phi :H^k(B;\mathbb{Z})\stackrel{\cong }{\to }{H}^{k+n}(\mathrm{Th}(\xi );\mathbb{Z}),\Phi (\alpha )={\pi }^{\ast }(\alpha )⌣u$$

where $\pi :D(\xi )\to B$ is the bundle projection and the cup product is taken in the relative cohomology $H^{k+n}(D(\xi ),S(\xi ))$.

Theorem (Thom isomorphism). For an oriented rank-$n$ vector bundle $\xi$ over a paracompact base $B$, the Thom class $u$ exists and is unique, and the map $\Phi$ is an isomorphism for all $k\ge 0$.

Key theorem with proof [Intermediate+]

Theorem (Existence and uniqueness of the Thom class). For an oriented rank-$n$ vector bundle $\xi$ over a paracompact base $B$, there exists a unique class $u\in H^n(D(\xi ),S(\xi );\mathbb{Z})$ restricting to the preferred generator on each fibre.

Proof. Existence. Use the Leray-Hirsch theorem. The key input is that the inclusion of each fibre $(D^n,S^{n-1})\hookrightarrow (D(\xi ),S(\xi ))$ induces an isomorphism $H^n(D(\xi ),S(\xi ))\to H^n(D^n,S^{n-1})\cong \mathbb{Z}$ when restricted to a contractible neighbourhood (local product structure). By the orientation hypothesis, these local generators agree on overlaps, giving a global class $u$ via a Mayer-Vietoris patching argument.

Uniqueness. The restriction to each fibre determines $u$ uniquely by the Leray-Hirsch theorem: $H^{\ast }(D(\xi ),S(\xi ))$ is a free $H^{\ast }(B)$-module with generator $u$, so any two Thom classes differ by multiplication by an element of $H^0(B)=\mathbb{Z}$; agreement on one fibre forces them to be equal. $\square$

Bridge. The Thom isomorphism translates the twisting of a vector bundle into a cohomological shift, generalising the suspension isomorphism of [topology.cw-complex] (which is the Thom isomorphism for the product bundle). The Thom class itself is a geometric refinement of the orientation data from [topology.singular-homology], and its square gives the Euler class, connecting to the Euler characteristic of [topology.euler-characteristic] via the Poincare-Hopf theorem.

Exercises [Intermediate+]

Advanced results [Master]

Thom spectrum. The assignment $B\mapsto \mathrm{Th}(\xi )$ for universal bundles produces the Thom spectrum $MO$ for unoriented cobordism, $MSO$ for oriented cobordism, and $MU$ for complex cobordism. Thom's 1954 theorem identifies the cobordism ring ${\Omega }_n^O$ with ${\pi }_n(MO)$. The Pontryagin-Thom construction provides the isomorphism ${\pi }_n(MO)\cong {\Omega }_n^O$ by sending a map $S^{n+k}\to \mathrm{Th}({\gamma }_k^O)$ to the preimage of the zero section, a closed $n$-manifold in $S^{n+k}$.

Stiefel-Whitney classes via Thom class. The mod 2 Thom class $\bar{u}\in H^n(\mathrm{Th}(\xi );\mathbb{Z}/2)$ exists for any real vector bundle (no orientability needed). The Stiefel-Whitney classes are defined by $w_i(\xi )={\Phi }^{-1}({\mathrm{Sq}}^i(\bar{u}))$, where ${\mathrm{Sq}}^i$ is the Steenrod square. This construction makes the Wu formula and the Whitney sum formula consequences of the Cartan formula for Steenrod squares.

Synthesis. The Thom isomorphism is the bridge between fibre-bundle geometry and cohomological algebra; it converts the oriented rank-$n$ bundle structure into an $n$-fold cohomological shift that generalises the suspension isomorphism from [topology.cw-complex], the Thom spectrum construction extends the CW-approximation machinery of 03.12.26 into the stable category where the Pontryagin-Thom isomorphism identifies homotopy groups with cobordism groups, and the fibre sequence of 03.12.28 applied to the sphere-bundle projection $S(\xi )\to B$ yields the Gysin sequence whose connecting map is cup product with the Euler class. The Thom class itself is the universal characteristic class, and all Stiefel-Whitney, Pontryagin, and Chern classes arise from its interaction with cohomology operations.

Full proof set [Master]

Proposition (Thom isomorphism via Serre spectral sequence). Let $\xi$ be an oriented rank-$n$ vector bundle over $B$ with projection $p:D(\xi )\to B$. The Serre spectral sequence for the pair $(D(\xi ),S(\xi ))$ collapses at $E_2$, and the Thom isomorphism is the identification $E_2^{k,0}=H^k(B)\cong E_2^{k,n}=H^{k+n}(D(\xi ),S(\xi ))$ given by cup product with $u$.

Proof. The spectral sequence for the relative cohomology of the disc-bundle pair has $E_2$ page $E_2^{p,q}=H^p(B;H^q(D^n,S^{n-1}))$. The local coefficient system $H^q(D^n,S^{n-1})$ is zero for $q\ne n$ and is the orientation local system $\mathbb{Z}$ for $q=n$ (by the orientability hypothesis). Hence $E_2^{p,q}=0$ for $q\ne 0,n$, so $d_r=0$ for $r\ge 2$ (the differentials map into or out of zero groups). The spectral sequence collapses at $E_2$, and the two surviving rows $q=0$ and $q=n$ give $H^k(D(\xi ))=E_{\infty }^{k,0}=H^k(B)$ and $H^{k+n}(D(\xi ),S(\xi ))=E_{\infty }^{k,n}=H^k(B;\mathbb{Z})\otimes \mathbb{Z}\cdot u$. The multiplicative structure identifies this tensor product as the cup product ${\pi }^{\ast }(\alpha )⌣u$. $\square$

Connections [Master]

Singular homology and cohomology [topology.singular-homology] provide the algebraic framework in which the Thom class lives; the Thom isomorphism is a cohomological tool that generalises the suspension isomorphism.

The Puppe fibre sequence 03.12.28 applied to the sphere-bundle projection $S(\xi )\to B$ produces the Gysin long exact sequence, connecting fibre-sequence technology to Thom-class technology.

Characteristic classes (Stiefel-Whitney, Pontryagin, Chern) arise from the Thom class via Steenrod operations and Chern-Weil theory, connecting the Thom isomorphism to the classification of vector bundles.

Bibliography [Master]

@article{thom1954,
  author = {Thom, Ren\'e},
  title = {Quelques propri\'et\'es globales des vari\'et\'es diff\'erentiables},
  journal = {Comment. Math. Helv.},
  volume = {28},
  pages = {17--86},
  year = {1954}
}

@book{milnor-stasheff,
  author = {Milnor, John W. and Stasheff, James D.},
  title = {Characteristic Classes},
  publisher = {Princeton University Press},
  year = {1974}
}

@book{bott-tu,
  author = {Bott, Raoul and Tu, Loring W.},
  title = {Differential Forms in Algebraic Topology},
  publisher = {Springer},
  year = {1982}
}

@book{hatcher2002,
  author = {Hatcher, Allen},
  title = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year = {2002}
}

Historical & philosophical context [Master]

Rene Thom introduced the Thom space and Thom isomorphism in his landmark 1954 paper "Quelques proprietes globales des varietes differentiables" ^{[Thom 1954]}, which also contains the cobordism theorem. Thom showed that the Thom space of the universal vector bundle classifies cobordism classes of manifolds, founding the field of cobordism theory.

The Thom isomorphism is philosophically significant because it shows that the topology of a vector bundle is completely encoded (cohomologically) by a single class and a shift in degree. This "local-to-global" principle -- a single class on each fibre extends uniquely to a global class that determines the entire cohomology of the Thom space -- is one of the deepest patterns in algebraic topology. The construction connects geometry (vector bundles, manifolds, embeddings) to algebra (characteristic classes, cohomology operations, spectral sequences) and underpins Thom's cobordism theory, which earned him the Fields Medal in 1958.