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For an $h$-oriented rank-$r$ real vector bundle, its Thom class restricts on each fiber to the suspension of the coefficient unit under the oriented identification $h^r(D^r,S^{r-1})\cong h^0(\mathrm{pt})$. Cup product with this class gives the Thom isomorphism. The associated Euler class in a generalized cohomology theory is the pullback along the zero section after forgetting relative supports.

// Target: geometry-and-topology.bigb

For a rank-$r$ real bundle with chosen stable complex normal orientation, the complex-cobordism Thom class gives the displayed isomorphism by $a\mapsto {\pi }^{\ast }a{u}_{\xi }$. Stabilizing by a trivial complex bundle suspends the Thom space and its class compatibly. The class of an actual complex rank-$n$ bundle has degree $2n$.

For an $R$-oriented rank-$d$ vector bundle, multiplication by the Thom class gives isomorphisms

For a real vector bundle over a compact base, its Thom space is the quotient of its disk bundle by its sphere bundle, with the collapsed sphere bundle as basepoint. Equivalently it is the one-point compactification of the total space. An oriented bundle's Thom class and Thom isomorphism identify the reduced cohomology of this space with shifted cohomology of the base.

For oriented real vector bundles $E$ and $F$, the ordered direct-sum orientation satisfiesThe formula follows because the Thom class of the direct sum is the fiberwise exterior product of the two Thom classes.