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 1 The vector bundle is orientable for a multiplicative cohomology theory if there exists an element which restricts to a generator on each fiberwise , where . Such a is called a Thom 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…)