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A simple proof of the result stated in the title is obtained by making the Casson-Gordon invariant additive. A. J. Casson and C. McA. Gordon proved in [1], [2] that there are algebraically slice knots which are not slice knots. In other words, the concordance group &, of algebraically slice knots is nontrivial. A natural question: Is dT infinitely generated? The author learned from Wu-chung Hsiang that Casson had obtained the affirmative answer to this question for some time; however, Casson has not published his proof to date. The purpose of this short note is to present a simple proof, which would possibly be different from Casson's. The main point in our argument is the observation that we can make the Casson-Gordon invariant additive by slightly generalizing its definition, thereby making it possible to detect linear independence in 9,. We shall use the language and notation of [2]. 1. Additivity of Casson-Gordon invariant for 3-manifolds. In [2], Casson and Gordon defined, for a closed oriented 3-manifold M and an epimorphism 4: H,(M) Z,,,, the invariant ar(M, 4), 0 M is the m-fold cyclic covering induced by 4. Pick up an m-fold cyclic branched covering of 4-manifolds W-> W, branched over a surface F c int W, such that a( W -> W) = (M -> M). (The existence of such (W, F) follows from Lemma 2.2 of [2].) Then define 2[F]2r(m r) ar(M, ) sign W er( W) -2 m For our purpose, we have to deal with arbitrary homomorphism 4. For simplicity, we shall restrict ourselves to the case m = p, a prime, so that a homomorphism H,(M) Zm is either epimorphic or trivial. DEFINITION. Let 4: H1(M) -> Zp be a homomorphism, where M is an oriented closed 3-manifold, p a prime. Define ar(M, )) as above if 4 is epimorphic, and define ar(M, 4) = 0 if 4 is trivial, for 0 < r <p. This invariant is additive in the following sense. Let M', M" be two closed Received by the editors June 30, 1980. 1980 Mathematics Subject Classification Primary 55A25; Secondary 57D90. 'This work was done in Fall, 1979, while the author was visiting the Institute for Advanced Study, Princeton, New Jersey, and partially supported by an NSF grant. The author is grateful to Professor W. C. Hsiang for helpful conversations. O 1981 American Mathematical Society 0002-9939/8 1 /0000-0442/$02.00

A simple proof that the concordance group of algebraically slice knots is infinitely generated