 He introduced the J-group "J"("X") of a finite complex "X", defined as the group of stable fiber homotopy equivalence classes of sphere bundles; this was later studied in detail by J. F. Adams in a series of papers, leading to the Adams conjecture.With Hirzebruch he extended the Grothendieck–Riemann–Roch theorem to complex analytic embeddings, and in a related paper  they showed that the Hodge conjecture for integral cohomology is false.The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.