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This paper gives some algebraic invariants for a piecewise linear imbedding of a surface into some 4-manifold inducing a Z or Q-homology isomorphism. Several examples are obtained by using these invariants. Let F be a closed (possibly disconnected) oriented surface and let W be a compact connected oriented piecewise linear 4-manifold with an isomorphism pq: Hq(F; R) H Hq(W; R) for all q > 0, where R = Z or Q, and such that the intersection number of any two elements of H2(W; Z)/(torsions) is 0. The purpose of this paper is to give some algebraic invariants which are necessary to find a piecewise linear imbedding F W inducing this isomorphism pq for all q > 0. Such invariants come from some developed arguments of quadratic forms of 3-manifolds defined by the author in [9]. By using these invariants, we shall have some examples. EXAMPLE 2.5. For each g > 1 there are compact connected orientable 4-manifolds W such that W is homotopy equivalent to a closed connected orientable surface Fg of genus g, but there is no (possibly nonlocally flat) piecewise linear imbedding from Fg to W inducing any homology isomorphism. For g = 1 this gives an elementary proof of a spineless 4-manifold announced by Y. Matsumoto [15]. In the higher even dimensional case, S. E. Cappell and J. L. Shaneson [1] and Y. Matsumoto [16] have constructed spineless manifolds W"n2 for each even n > 4 whose statements are weaker than the above. EXAMPLE 2.6. For each g > 0 there are infinitely many relatively nonhomology cobordant compact connected orientable 4-manifolds W such that there is a homotopy equivalent piecewise linear imbedding Fg W, but there is no locally flat piecewise linear imbedding from Fg to W inducing any homology isomorphism. Let L be a link of s components with linking numbers 0 (i.e. any two components of L have the linking number 0) and let WL be a 4-manifold obtained from a 4-cell D4 by attaching s 2-handles along the link L in S3 = aD4 with null-homologous framings. Clearly, WL is homotopy equivalent to a bouquet S2 V S2 V... VS2 of s 2-spheres and any two elements of H2(WL; Z) have the Received by the editors November 15, 1978 and, in revised form, October 9, 1979. AMS (MOS) subject classifications (1970). Primary 57C35; Secondary 55A10, 57C20.

On a 4-manifold homology equivalent to a bouquet of surfaces