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1004_(GTM170)Sheaf Theory | 0 |
# GraduateTexts inMathematics
Glen E. Bredon
# Sheaf Theory
Second Edition Springer
Editorial Board
S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3... |
1004_(GTM170)Sheaf Theory | 1 | ology theories.
The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Alge... |
1004_(GTM170)Sheaf Theory | 2 | this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the autho... |
1004_(GTM170)Sheaf Theory | 3 | m of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) .
Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit
\[
{\mathcal{A}}... |
1004_(GTM170)Sheaf Theory | 4 | eq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mat... |
1004_(GTM170)Sheaf Theory | 5 | ight| = \{ x \in X \mid s\left( x\right) \neq 0\} \) .
The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 .
1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -man... |
1004_(GTM170)Sheaf Theory | 6 | thcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V... |
1004_(GTM170)Sheaf Theory | 7 | cal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}}
\]
\[
{\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}}
\]
\[
\text{...}
\]
\( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal... |
1004_(GTM170)Sheaf Theory | 8 | presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions.
4.2. Definition. If \( \mathcal{... |
1004_(GTM170)Sheaf Theory | 9 | sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -c... |
1004_(GTM170)Sheaf Theory | 10 | Y \) . Similarly, the total torsion product is defined to be
\[
\mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right)
\]
Clearly, we have natural isomorphisms
\[
{\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle ... |
1004_(GTM170)Sheaf Theory | 11 | ns over nonopen subspaces, and in particular that
\[
\operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right)
\]
in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack ... |
1004_(GTM170)Sheaf Theory | 12 | and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \)
If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while ... |
1004_(GTM170)Sheaf Theory | 13 | ups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology.
## Singular cohomology
Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \)... |
1004_(GTM170)Sheaf Theory | 14 | ightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by
\[
{Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldo... |
1004_(GTM170)Sheaf Theory | 15 | \mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories.
## Exercises
1. (c) If \( \mathcal{A} \) is a sheaf on... |
1004_(GTM170)Sheaf Theory | 16 | \left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \)
16. Show that there are natural isomorphisms
\[
\operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right)
\]
a... |
1004_(GTM170)Sheaf Theory | 17 | {2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \)
1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters).
We have th... |
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