Pin and Pin′ cobordism

The cobordism group QP*i'of smooth manifolds with a Pin structure on the stable normal bundle is computed. The image Qpi'-9* is determined, and some generators for QPi and Qi"' are given. 1. Notation. All manifolds will be smooth of class C', and compact. The Lie group Pin(n) is the double covering of O(n) whose identity component is Spin(n). (See [3] for details.) Let 0, SO, Spin, and Pin denote the usual stable groups. A bundle $ has a Pin structure if w2(5)+w w(5)=0. To see this consider the diagram of fibrations B Pin(1)B Pin BO(1) >BOf K(Z2, 2). We need to findf*(t). But since the identity component of B Pin is B Spin, f*(l) =W2+aivi where a E Z2. Since Pin(1)=Z4, the fibration B Pin(1)-* BO(1) is nontrivial, hence i*f *(t)?O. But w2=O in BO(1), so a=1. (This argument is due to R. E. Stong, and corrects the statement in [6].) A manifold is called a Pin mainfold if its stable tangent bundle 7 has a Pin structure, and a Pin' manifold if its stable normal bundle v has a Pin structure. Pin and Pin' do not coincide, since w2(T)=w2(v)+w)(v). For example, RP4n is a Pin' manifold, and RP4n+2 is a Pin manifold. In [2] Q"I was calulated by using the isomorphism Q`1,-QsP'11(RP). Analogously there is the following long exact sequence, due to Stong, relating Spin and Pin' cobordism, where A is the cobordism theory introduced in [4] *QSpin An-> Q P i n' -> Q S pi n 2. Algebraic methods. Let M be a closed manifold. Since the stable normal bundle v of M has a Pin structure iff w2(v)+ w(v) =, the classifying space B Pin' may be obtained from BO by killing W2+ w1, i.e., B Pin'(n) is the total space of the fibration over BO(n) induced from the path space Received by the editors August 8, 1972 and, in revised form, September 27, 1972. AMS (MOS) subject classifications (1970). Primary 57D90; Secondary 55G10, 55H15.