The first counterexample was constructed by . Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class such that for some positive integer —which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class. reinterpreted their result in the framework of cobordism and found many examples of such classes. The simplest adjustment of the integral Hodge conjecture is:  Equivalently, after dividing formula_45 by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle. This is also false. found an example of a Hodge class which is not algebraic, but which has an integral multiple which is algebraic. A natural generalization of the Hodge conjecture would ask:  This is too optimistic, because there are not enough subvarieties to make this work. A possible substitute is to ask instead one of the two following questions:  Hodge made an additional, stronger conjecture than the integral Hodge conjecture. Say that a cohomology class on "X" is of "co-level c" (coniveau c) if it is the pushforward of a cohomology class on a "c"-codimensional subvariety of "X". The cohomology classes of co-level at least "c" filter the cohomology of "X", and it is easy to see that the "c"th step of the filtration "N'H"("X", Z) satisfies  Hodge's original statement was: This version is open. The strongest evidence in favor of the Hodge conjecture is the algebraicity result of . Suppose that we vary the complex structure of "X" over a simply connected base. Then the topological cohomology of "X" does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations. Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.