Lectures On Sheaf Theory By C.h. Dowker

Lectures on Sheaf Theory by C.H. Dowker Tata Institute of Fundamental Research Bombay 1957 Lectures on Sheaf Theory by C.H. Dowker Notes by S.V. Adavi and N. Ramabhadran Tata Institute of Fundamental Research Bombay 1956 Contents 1 Lecture 1 1 2 Lecture 2 5 3 Lecture 3 9 4 Lecture 4 15 5 Lecture 5 21 6 Lecture 6 27 7 Lecture 7 31 8 Lecture 8 35 9 Lecture 9 41 10 Lecture 10 47 11 Lecture 11 55 12 Lecture 12 59 13 Lecture 13 65 14 Lecture 14 73 iii iv 15 Lecture 15 16 Lecture 16 17 Lecture 17 18 Lecture 18 19 Lecture 19 20 Lecture 20 21 Lecture 21 22 Lecture 22 23 Lecture 23 24 Lecture 24 25 Lecture 25 26 Lecture 26 27 Lecture 27 28 Lecture 28 29 Lecture 29 30 Lecture 30 31 Lecture 31 32 Lecture 32 33 Lecture 33 Contents 81 87 93 101 107 113 123 129 135 139 143 147 155 161 167 171 177 183 189 Lecture 1 Sheaves. 1 Definition. A sheaf S = (S,τ,X) of abelian groups is a map π : S −−−→ X, where S and X are topological spaces, such that 1. π is a local homeomorphism, 2. for each x ∈ X, π−1(x) is an abelian group, 3. addition is continuous. That π is a local homeomorphism means that for each point p ∈ S, there is an open set G with p ∈ G such that π|G maps G homeomorphi-cally onto some open set π(G). Sheaves were originally introduced by Leray in Comptes Rendus 222(1946)p. 1366 and the modified definition of sheaves now used was given by Lazard, and appeared first in the Cartan Sem. 1950-51 Expose 14. Inthe definition ofasheaf, X isnot assumed tosatisfy anyseparation axioms. S is called the sheaf space, π the projection map, and X the base space. The open sets of S which project homeomorphically onto open sets of X form a base for the open sets of S. Proof. If p is in an open set H, there exists an open G, p ∈ G such that π|G maps G homeomorphically onto an open set π(G). Then H ∩ G is open, p ∈ H∩G ⊂ H, and η|H∩G maps H∩G homeomorphically onto π(H ∩G) open in π(G), hence open in X. 1 ... - tailieumienphi.vn