4 - manifolds with free fundamental group 419

for any x,yeH^ {D, A). Here we identify H^ (D, A) with H^ {X, A) via/*. Summarizing, the following result holds.

Lemma 5. Letf : X^ -^ Dbe two polarized 4-dimensional Poincaré spaces over a 2-stage Postnikov system DJor i = 1,2. Then X^ andX2 have the same inter section form if and only if

3 . Poincaré spaces X with free fundamental group and H2 {X, Q) Ф 0

In this section we will prove Theorem 1 for oriented Poincaré spaces X with Щ{Х) ^ *p/ and H2{X, Q) Ф 0.

Because U^{X) is a free group, the ring A = Т\П^ is hereditary. Because the augmentation ideal is projective, A has cohomological dimension < 1. Then the usual universal coefficient theorems hold (see for example [4], p. 114), hence H2 {X, A) and H^ (Z, A) are free Л-modules. Indeed, Ext (Я2 {X, Л), A) injects into H^ (X, A) = H^ {X, A) = 0. Since A is not Noetherian, one needs to see why H2 {X, A) is finitely generated. It follows from the spectral sequence of the covering X -> Хгпа of ЯзСБЯ^, Z) = 0 that H2{X, Z) = H2IX, A) ®^ Z. Hence, if ЯзСХ, Z) = Z\ then wehaveЯ2(Z,Л) = Л^

According to section 2 we have to prove the following two propositions.

Proposition 6. Let П be a finitely generated free group and let X^, X2 be oriented connected 4-dimensional Poincaré spaces with fundamental group П. If)^xx ^^ isometric to kx^, then there exist D-polarizations f \ X^ -^ D such that

Proposition 7. The homomorphism

F : ЯДДZ ) -^ Нотл-л(ЯЧД^)®1А'(А A^) is injective.

Proof of Proposition 6. Let

cf> : H2 { X , , Ä ) - ^H2 { X2 . A ) and

хр : П , { Х , ) ^П , { Х2 )

be an isomorphism between {H2{X^, Л), Я^) and (Я2(Х2, Л), Я2). Note that

П2 { Х , ) с^Н2 { Х , , А ) ^Н2 { Хд

and D^ = К{П2{Х^), 2). Let/;. : X^ -> Д be a 3-equivalence, for / = 1, 2.