Involutions on Spin 4-manifolds

We show that a simply-connected spin 4-manifold which admits a locally linear involution must have vanishing signature. We also show that the codimensions of all components of the fixed point set of an involution on a spin 4-manifold are the same modulo 4 . There is no assumption of local linearity in this result, which extends a lemma of Atiyah and Bott. This note records two results about involutions on spin 4-manifolds. The first result is an extension to the case of arbitrary involutions of a well-known lemma of Atiyah and Bott [AB]. Their lemma, proven in the smooth case in [AB], and extended to the locally linear case in [E2], states that the codimensions of all components of the fixed point set must agree modulo 4. The second result is a theorem which states that a simply-connected spin 4-manifold which admits a locally linear involution acting trivially on the homology must have vanishing signature. This was inspired by the fact that a holomorphic Z 2-action on a K3 surface cannot be homologically trivial. Edmonds [El] has constructed locally linear, homologically trivial actions of Zp for p an odd prime on virtually all simply connected 4-manifolds. 1. Codimensions of fixed points Theorem 1.1. Let t : X -> X be an involution on the spin 4-manifold X which preserves orientation and a spin structure on X. Then the codimensions of all components of the fixed point set of x agree modulo 4. The outline of this Atiyah-Bott lemma in the smooth case is as follows: A spin structure a is a double covering P -+ P of the principal frame bundle P restricting to a non-trivial cover in each fiber; we say that t preserves a if it lifts to a fiber preserving map f on this double cover. In each fiber Px , t has order 1 or 2 ; correspondingly, we say that x has odd or even type. It is easy to see that the type must be the same in every fiber. Now, on the spheres linking the components of the fixed point set, x is the antipodal map and hence has odd type if the codimension is 2 mod 4 and even type if the codimension is 0 mod 4. Thus the result follows. Received by the editors April 25, 1993. 1991 Mathematics Subject Classification. Primary 57S17, 57N13. Author partially supported by an NSF Postdoctoral Research Fellowship. © 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page