H. CARTAN and J.-P. SERRE have shown how fundamental theoremson holomorphically complete manifolds （STEIN manifolds） can be for-mulated in terms of sheaf theory. These theorems imply many facts offunction theory because the domains of holomorphy are holomorphicallycomplete. They can also be applied to algebraic geometry because thecomplement of a hyperplane section of an algebraic manifold is holo-morphically complete. J.-P. SERRE has obtained important results onalgebraic manifolds by these and

Introduction
Chapter One. Preparatory material
1. Multiplicative sequences
2. Sheaves
3. Fibre bundles
4. Characteristic classes
Chapter Two. The cobordism ring
5. PONTRJAGIN numbers
6. The ring
7. The cobordism ring
8. The index of a 4 k-dimensional manifold
9. The virtual index
Chapter Three. The TODD genus
10. Definition of the TODD genus
11. The virtual generalised TODD genus
12. The T-characteristic of a G L (q， C)-bundle
13. Split manifolds and splitting methods
14. Multiplicative properties of the TODD genus
Chapter Four. The RIEMANN-ROCH theorem for algebraic manifolds
15. Cohomology of compact complex manifolds
16. Further properties of the Xy-characteristic
17. The virtual Xy-characteristic
18. Some fundamental theorems of KODAIRA
19. The virtual Xy-characteristic for algebraic manifolds
20. The RIEMANN-ROCH theorem for algebraic manifolds and complex analytic line bundles
21. The RIEMANN-ROCH theorem for algebraic manifolds and complex analytic vector bundles
Appendix One by R. L. E. SCHWARZENBERGER
22. Applications of the RIEMANN-ROCH theorem
23. The RIEMANN-ROCH theorem of GROTHENDIECK
24. The GROTHENDIECK ring of continuous vector bundles
25. The ATIYAH-SINGER index theorem
26. Integrality theorems for differentiable manifolds
Appendix Two by A. BOREL
A spectral sequence for complex analytic bundles
Bibliography
Index