Pin Structures on Low–dimensional Manifolds

Pin structures on vector bundles are the natural generalization of Spin structures to the case of nonoriented bundles. Spin(n) is the central Z/2Z extension (or double cover) of SO(n) and Pin−(n) and Pin(n) are two different central extensions of O(n), although they are topologically the same. The obstruction to putting a Spin structure on a bundle ξ (= R → E → B) is w2(ξ) H(B;Z/2Z); for Pin it is still w2(ξ), and for Pin − it is w2(ξ) + w 1(ξ). In all three cases, the set of structures on ξ is acted on by H(B;Z/2Z) and if we choose a structure, this choice and the action sets up a one–to–one correspondence between the set of structures and the cohomology group. Perhaps the most useful characterization (Lemma 1.7) of Pin± structures is that Pin− structures on ξ correspond to Spin structures on ξ ⊕ det ξ and Pin to Spin structures on ξ ⊕ 3 det ξ where det ξ is the determinant line bundle. This is useful for a variety of “descent” theorems of the type: a Pin± structure on ξ ⊕ η descends to a Pin (or Pin− or Spin) structure on ξ when dim η = 1 or 2 and various conditions on η are satisfied. For example, if η is a trivialized line bundle, then Pin± structures descend to ξ (Corollary 1.12), which enables us to define Pin± bordism groups. In the Spin case, Spin structures on two of ξ, η and ξ ⊕ η determine a Spin structure on the third. This fails, for example, for Pin− structures on η and ξ ⊕ η and ξ orientable, but versions of it hold in some cases (Corollary 1.15), adding to the intricacies of the subject. Another kind of descent theorem puts a Pin± structure on a submanifold which is dual to a characteristic class. Thus, if V m−1 is dual to w1(TM ) and M is Pin±, then V ∩| V gets a Pin± structure and we have a homomorphism of bordism groups (Theorem 2.5),