Abstract: This paper proposes a framework within which to classify pairs (M n, [ , ]), M ~ being a closed differentiable manifold and [ , ]:Hk(M; Zz) ®z2 Hk(M; Z2)-~ Z2 being a symmetric bilinear form on the k-th mod2 cohomology group of M. This problem arose in studying the work of Lusztig, Milnor, and Peterson [2] on semicharacteristics. For a closed oriented manifold M2r+ 1 of dimension 2r + 1 they examine the form on H'(M; Z2) defined by [x, y] = (xwSqly, [M]). On the projective space RP(3), this form is of rank one, but RP(3) is parallelizable, hence bounds. Thus the rank of the form is not an invariant of the cobordism class, and one needs some finer structure in order to analyze this form. In the process, one encounters the problem of finding those subsets of H*(M~; Z2) which can be the image ofH*(V n+l ; Z2) for some compact manifold with boundary V with M n a submanifold of the boundary of V. Some partial characterizations are obtained.