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Abstract. We introduce a gauge-theoretic integer valued lift of the Rohlin invariant of a smooth 4-manifold X with the homology of S 1 × S 3 . The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a generic 1-parameter family.

Knots in S 3 can be decomposed into simpler pieces in several different ways. The most basic is by connected sum into prime pieces; such a decomposition is unique. Further, it is easy to understand the contribution of each summand to any knot invariant one might wish to compute. Another splitting of knots occurs when a 2-sphere (called a Conway sphere) in S 3 hits the knot transversally in four points. The resulting splitting into so-called tangles has proved quite fruitful in various investigations [7,8,22,6] of symmetries and other properties of knots. Given a Conway sphere, there is an operation called mutation which yields a new knot. Roughly speaking, one takes out the tangle, flips it over and glues it back in. The resulting knot tends to differ from the original one, unless the tangle on one side was symmetric [6]. However, many invariants of a knot are preserved by mutation [7], e.g. the signature and Alexander polynomial, as well as the new two-variable knot polynomial [12].In this paper we show that the Gromov norm (see below) of a knot and its mutant coincide. In particular if S3--K is a hyperbolic manifold then S3-mutant of K is as well, and their volumes are the same. These results are instances of a more general theorem which shows that Gromov's norm is preserved by certain kinds of cutting and pasting along surfaces. By using the torus decomposition of a general 3-manifold, one reduces the problem to understanding what happens for hyperbolic manifolds. For the case of a hyperbolic manifold we show that for certain surfaces F c M and symmetries r of F, cutting M along F and regluing via z results in a new hyperbolic manifold M' with vol(M)--vol(M'). Colin Adams [1] has proved a similar result for the special case of a thrice-punctured sphere in M. Our basic method can be extended to show that for Dehn surgeries on a knot which produce hyperbolic manifolds, the volume is the same for the corresponding surgeries on the mutant knot. We also give a similar result about the volumes of the branched cyclic covers of the knot and its mutant. In a future paper, we will consider the effect of this sort of cutting and pasting on the Chern-Simons invariant and ~/-invariant of hyperbolic manifolds. In the course of the proof of the theorem concerning hyperbolic manifolds, we need to use the results of [I1] on embeddedness and intersections of least area surfaces. The theorems of [11] deal only with the compact case, and we need their analogues in the case of finite-volume hyperbolic 3-manifolds. These

Let C T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C ∆ be the subgroup generated by knots with trivial Alexander polynomial. We prove C T /C ∆ is infinitely generated. Our methods reveal a similar structure in the 3-dimensional rational spin bordism group, and lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.

We study the space of positive scalar curvature (psc) metrics on a 4-manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1-parameter Seiberg-Witten invariants which we introduced in earlier work. The invariant shows that the diffeomorphism group of the underlying 4-manifold is disconnected. We also study the moduli space of positive scalar curvature metrics modulo diffeomorphism, and give examples to show that this space can be disconnected. The (non-orientable) 4-manifolds in this case are explicitly described, and the components in the moduli space are distinguished by a P in c eta invariant.