Let be a multiplicative cohomology theory. We say that is **complex-oriented** if one is given the data of an element which restricts to the canonical generator of . It turns out that one has a bit more: a complex orientation gives on a functorial, multiplicative choice of *Thom classes* for complex vector bundles. In fact, this is a perhaps more natural definition of such a theory.

What does this mean? Given a vector bundle , one can form the Thom space : in other words, the quotient of the unit ball bundle in (with respect to a choice of metric) by the unit sphere bundle . When is compact, this is just the one-point compactification of .

Definition 1The vector bundle isorientablefor a multiplicative cohomology theory if there exists an element which restricts to a generator on each fiberwise , where . Such a is called aThom class.

Observe that for each point , there is a restriction map if the dimension of is .

The existence of a Thom class implies a Thom isomorphism, as for ordinary homology.

Theorem 2 (Thom isomorphism)A Thom class induces an isomorphism

given by cup-product with .

In the case of ordinary homology, a Thom class is unique (up to sign) if it exists; in general, though, a Thom class is highly non-unique, and an orientation is *additional data *than simple orientability.

Here are a few basic cases:

- Any vector bundle is orientable for -cohomology.
- An
*oriented*(in the usual sense: i.e., the top wedge power is trivial) vector bundle is one oriented for -cohomology. - Complex vector bundles are oriented for -theory. We will see this below.
- Spin bundles are oriented for -theory. An explicit construction of Thom classes can be made, as virtual bundles arising from Clifford modules: this is in Atiyah-Bott-Shapiro’s paper.
- A trivial bundle is orientable for any cohomology theory (this is rather uninteresting: the Thom space is just a suspension). (more…)