Orientability of Ptxed Point Sets

It is proved that the fixed point set of a smooth involution which preserves orientation and a spin structure on a smooth manifold is necessarily orientable. As an application it is proved that a simply connected spin 4-mamfold with nonzero signature admits no involution which acts by multiplication by -1 on its second rational homology group. Consider a smooth orientation-preserving action of the cyclic group of order 2, Z2, on a manifold M. As is well known, in contrast to the case of odd order group actions, the fixed point set F is a manifold which need not be orientable. The simplest example is that of complex conjugation on CP2 which fixes RP2. In this note we shall show that if in addition the action preserves a spin structure on M, then F is necessarily orientable. This result will then be applied to show that a closed simply connected spin 4-manifold A/4 of nonzero signature does not support a smooth Z2 action which acts by multiplication by -1 on H2(M; Q). The crux of the argument is in the following proposition about Z2 vector bundles over the 2-sphere. Proposition 1. Let £ be a k-dimensional Z2 vector bundle over S2, where Z2 acts on S2 by reflection through the equator Sx. Then the fixed subbundle (i\Sx)Zl is orientable if and only if £ is stably trivial as a vector bundle. Proof. If dim(£| S x)z* = dim(£| S '), then any nonequivariant trivialization of £\D2, where Z>2 is the upper hemisphere of S2, extends to an equivariant trivialization of £. So in this case both £ and (£\Sx)z* are trivial. Next consider the case when £ is 2-dimensional and (i-\Sx)Z2 is 1-dimensional. Let S2 c E(£) denote the zero section. Then £ is stably trivial if and only if the self-intersection number S2 • S2 = 0 (mod 2). To compute S2S2 we carefully perturb S2 to a section S2 transverse to S2 and count S2 n S2. Suppose that (£|S')Z2 is trivial. First perturb the equator Sx in £(£|S')Z2 to a nonvanishing section Sx. This extends to some section D2 of £|Z>2, transverse to Z>2. By equivariance this extends to a section D2 of £|£)2, transverse to D2. Then 52 = ß2ui>2isa topological section transverse to S2. Clearly S2 n S2 consists of an even number of points. Now suppose (£|S')Z2 is nontrivial. Then we may choose a section Sx of (£\SX)Z2 which vanishes at exactly one point of transverse intersection with the zero section Sx. This section extends to a section D2 of £|D2 whose interior meets D\ in Received by the editors July 16, 1980. AMS (MOS) subject classifications (1970). Primary 57E15, 55C20; Secondary 57D15. 'Supported in part by a National Science Foundation grant. 120 © 1981 American Mathematical Society 0002-9939/81 /0000-0224/$02.2 5 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ORIENTABILITY OF FIXED POINT SETS 121 finitely many points of transverse intersection. By equivariance extend this to a topological section S2 of £. Clearly S2 n S2 consists of an odd number of points. It remains only to observe that S2 is topologically transverse to S2. To see this consider a standard model for a neighborhood of {i} = S'nS' in £(£), as follows: Coordinatize a neighborhood by C X C, where the first factor is the base and the second factor is the fiber, and Z2 acts by complex conjugation in each factor. Then Sx corresponds to R X 0 and (Ç\SX)Z2 corresponds to R X R. Simply perturb R X 0 to R = {(x, x): x G R}. This may be accomplished by perturbing C X 0 to C = {(z, z): z G C}. Then clearly R is transverse to R X 0 in R X R and C is transverse to C X 0 in C X C. Since the perturbation S2 of S2 above may be chosen this way near S ' n Sx, we see that S2 is indeed transverse to S2 as claimed. It remains to reduce the general case to the cases already considered. First of all, we may assume (£|S')Z2 has positive dimension by adding to £ a trivial bundle S2 X R with trivial Z2 action in the fiber, if necessary. Clearly this does not affect the orientability of (£\Sx)Zl or the stable triviality of £. Decompose £|S'1 = ££, © £+, into eigenbundles for the eigenvalues ± 1, where r + s = k. We have arranged that s > 1. The first case considered was when r = 0; so we may also assume r > \. The two eigenbundles can be destabilized to line bundles: ^.«e:,-1©^, and £'+1 « e°~xx © {',. Let £q2 = £_*i © £|i and e = er_\~x © es~tx. We claim that the equivariant destabilization £|S,|=£02ffie*~2 extends to an equivariant destabilization of £ over S2,