LIFTS OF SMOOTH GROUP ACTIONS TO LINE BUNDLES

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.

SMOOTH GROUP ACTIONS ON 4-MANIFOLDS AND SEIBERG–WITTEN INVARIANTS

In this paper we study the Seiberg–Witten invariants of 4-manifold acted on by a finite group (or a compact Lie group). Among other things, we have: Let X be a smooth closed 4-dimensional ℤp-manifold, where p is a prime. Suppose H1(X,ℝ) = 0 and [Formula: see text] where [Formula: see text]. If ℤp acts trivially on the space [Formula: see text] of self dual harmonic 2-forms, then, for any ℤp-equivariant Spin c-structure [Formula: see text] on X, the Seiberg–Witten invariant satisfies [Formula: see text] provided [Formula: see text] for j = 0,1,…,p-1, where [Formula: see text], DA: Γ(W+) → Γ (W-) is the equivariant Dirac operator determined by [Formula: see text] for an equivariant connection A on det W+.