Sheaf Theory

I. Sheaves and presheaves
II. Sheaf cohomology
III. Comparison with other cohomology theories
IV. Application of spectral sequences
V. Borel-Moore homology
VI. Cosheaves and cech homology

A. Spectral sequences
B. Solutions to selected exerciese

This book emphasizes the role of sheaves in defining and comparing various cohomology theories, particularly in algebraic topology. Key innovations in the book include the introduction and use of the concept of “tautness” of a subspace, proof that sheaf-theoretic cohomology satisfies the homotopy property for general topological spaces, and the incorporation of relative cohomology into sheaf theory. While the book focuses on algebraic topology, it also touches on applications in other mathematical fields, though it does not delve deeply into algebraic geometry.