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We prove a result which supports the Waldhausen Conjecture, i.e., suppose M is an irreducible orientable 3-manifold with 17r1 (M)I = c; if M admits an orientation reversing involution t, and M has a nontrivial finite cover, then some finite cover M of M has positive first Betti number. An important conjecture in 3-manifolds theory is the so-called Waldhausen Conjecture. Suppose M is an irreducible orientable 3-manifold with K7 (M)I = oo, then M is finitely covered by a Haken manifold. Waldhausen Conjecture is a corollary of the so-called Strong Waldhausen Conjecture. Suppose M is an orientable 3-manifold with 17 (M)I = ox, then M is finitely covered by some 3-manifold with positive first Betti number. Remark. Any closed nonorientable 3-manifold has positive first Betti number (see [2]). In this note we prove the following theorem which supports the Strong Waudhausen Conjecture. Theorem. Suppose M is an irreducible orientable 3-manifold with 7 I(M) = oo. If M admits an orientation reversing involution T, and M has a nontrivial finite cover, them some finite cover M of M has positive first Betti number. A rational homology 3-sphere is a 3-manifold M such that H*(M; Q) = H*(S3; Q); here Q is the field of rational numbers and S3 is the 3-sphere. Since we only consider compact 3-manifolds in this note, the fact that the first Betti number of a closed 3-manifold M is zero is equivalent to the fact that M is a rational homology 3-sphere. We say a finite group G acts freely on M, if for any g E G, fix(g) $& 0 implies that g = id; here fix(g) is the fixed point set of g . Proof of theorem. Suppose a closed irreducible orientable 3-manifold M is not S3 or RP3, and M admits an orientation reversing involution T such that Received by the editors January 18, 1988 and, in revised form, August 5, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N10. (? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

The virtual 𝑍-representability of 3-manifolds which admit orientation reversing involutions