Rational homology cobordisms of spherical space forms

Abstract

THE RECENT work of Donaldson [S-7] and of the authors [8-lo] has pointed out that the study of the differential geometric and analytic nature of a smooth 4-manifold in the guise of the study of Yang-Mills connections yields new and surprising results. This is surely no accident. In this paper we will begin to place under the umbrella of Yang-Mills theory many of the earlier results in 4-manifolds which utilize invariants arising from the G-signature theorem and we will generalize these results. Our main result concerns rational homology cobordisms of spherical space forms. Let W’ be a compact smooth 4-manifold with boundary components d,, . . , Zp which are spherical space forms and suppose that W4 has the rational homology of a p-punctured sphere. To any character x: H,( W4; Z)U(1) we associate three integers G( W4, ,Y), p( If’, x) and p( W4, x) as follows. Each boundary component dj of W4 is a quotient of S3 by a finite group Gj acting orthogonally on S’ and which extends to an orthogonal action on D’ fixing the origin. For g E Gj, let rj(g) and sj(g) denote the rotation angles of the action of g on D’. Let xj= d*: H,(d,; Z)U( 1) wherej is the inclusion of d, into WA. The character 1 determines a flat SO(2) bundle L, over W4 which when restricted to a boundary component Zj is the SO(2) bundle S3 x S’/Gj+S3/Gj