Compactly supported de Rham cohomology - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. Because the exterior derivative does not enlarge support, the spaces of compactly supported differential forms form the subcomplex

0\longrightarrow\Omega_c^0(M)\xrightarrow{d}\Omega_c^1(M)\xrightarrow{d}\cdots

of the de Rham complex. The compactly supported de Rham cohomology of $M$ is

H_{c,\mathrm{dR}}^k(M) =\frac{\ker(d:\Omega_c^k(M)\to\Omega_c^{k+1}(M))} {\operatorname{im}(d:\Omega_c^{k-1}(M)\to\Omega_c^k(M))}.

Thus a class is represented by a closed compactly supported $k$ -form, and two representatives agree when their difference is the exterior derivative of a compactly supported $(k-1)$ -form.

Functoriality and support

If $f:M\to N$ is a proper smooth map, pullback preserves compact support and induces $f^*:H_{c,\mathrm{dR}}^k(N)\to H_{c,\mathrm{dR}}^k(M)$ . Arbitrary smooth maps need not do so. If $j:U\hookrightarrow M$ is an open inclusion, extension by zero sends compactly supported forms on $U$ to compactly supported forms on $M$ ; smoothness holds because each support is closed away from the boundary of $U$ .

Duality and examples

For an oriented $n$ -manifold, wedge product followed by integration gives the Poincaré pairing

H_{c,\mathrm{dR}}^k(M)\times H_{\mathrm{dR}}^{n-k}(M)\longrightarrow\mathbb R.

Its nondegeneracy is the compact-support form of Poincaré duality Bott and Tu, Chapter I. If $M$ is compact, compactly supported and ordinary de Rham cohomology coincide. For $\mathbb R^n$ , only the top-degree compactly supported group is nonzero, and integration identifies it with $\mathbb R$ .

References

Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter I, compact supports and Poincaré duality.
Glen E. Bredon, Topology and Geometry, Springer, 1993. DOI record. Relevant: Chapter VI, de Rham theory and compact supports.