Alexander-Spanier cohomology: Information from Answers.com

In mathematics, particularly in algebraic topology Alexander-Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. It is similar to and in some sense dual to de Rham cohomology, and in the context of de Rham cohomology is often called cohomology with compact support. It is named for J. W. Alexander and Edwin Henry Spanier (1921-1996).

Warning:In fact, the usual terminology for what is described below seems to be " de Rham cohomology with compact support″ or "compactly supported de Rham cohomology″, for example in the main reference Bott-Tu. In many mathematical texts the term "Alexander-Spanier cohomology″ theory is used for a different cohomology theory, which is defined for arbitrary topological spaces.

Given a manifold X, let $\Omega^k_{\mathrm c}(X)$ be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative. Then the Alexander-Spanier cohomology groups $H^q_{\mathrm c}(X)$ are the homology of the chain complex $(\Omega^\bullet_{\mathrm c}(X),d)$ :

$0 \to \Omega^0_{\mathrm c}(X) \to \Omega^1_{\mathrm c}(X) \to \Omega^2_{\mathrm c}(X) \to \cdots$

i.e., $H^q_{\mathrm c}(X)$ is the vector space of closed q-forms modulo that of exact q-forms.

Despite their definition as the homology of an ascending complex, the Alexander-Spanier groups demonstrate covariant behavior; for example, given the inclusion mapping j for an open set U of X, extension of forms on U to X (by defining them to be 0 on X–U) is a map $j_*: \Omega^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X)$ inducing a map

$j_*: H^q_{\mathrm c}(U) \to H^q_{\mathrm c}(X)$ .

They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let f: Y → X be such a map; then the pullback

$f^*: \Omega^q_{\mathrm c}(X) \to \Omega^q_{\mathrm c}(Y): \sum_I g_I \, dx_{i_1} \wedge \ldots \wedge dx_{i_q} \mapsto (g \circ f) \, d(x_{i_1} \circ f) \wedge \ldots \wedge d(x_{i_q} \circ f)$

induces a map

$H^q_{\mathrm c}(X) \to H^q_{\mathrm c}(Y)$ .

If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence

$\cdots \to H^q_{\mathrm c}(U) \overset{j_*}{\longrightarrow} H^q_{\mathrm c}(X) \overset{i^*}{\longrightarrow} H^q_{\mathrm c}(Z) \overset{\delta}{\longrightarrow} H^{q+1}_{\mathrm c}(U) \to \cdots$

called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X.

Alexander-Spanier cohomology satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then

$\cdots \to H^q_{\mathrm c}(U \cap V) \to H^q_{\mathrm c}(U)\oplus H^q_{\mathrm c}(V) \to H^q_{\mathrm c}(X) \overset{\delta}{\longrightarrow} H^{q+1}_{\mathrm c}(U\cap V) \to \cdots$

where all maps are induced by extension by zero is also exact.

References

Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3

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