{"id":2045,"date":"2012-02-26T20:20:29","date_gmt":"2012-02-26T12:20:29","guid":{"rendered":"http:\/\/wamath.sinaapp.com\/?p=2045"},"modified":"2012-02-26T20:20:29","modified_gmt":"2012-02-26T12:20:29","slug":"review-of-the-rham-cohomology-theory","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2045","title":{"rendered":"Review of the Rham Cohomology Theory"},"content":{"rendered":"

Let $M$ be a smooth closed (i.e., compact and without boundary) manifold. Let $TM$ and $T^\\ast M$ denote the tangent and cotangent (vector) bundle of $M$, respectively. Denote $\\Lambda^\\cdot(T^\\ast M)$ the complex exterior algebra bundle of $T^\\ast M$ and $\\Omega^\\cdot(M,\\C)=\\Gamma(\\Lambda^\\cdot(T^\\ast M))$ be the space of smooth sections of $\\Lambda^\\cdot(T^\\ast M)$. Particularly, for any integer $p$ with $0\\leq p\\leq n=\\dim M$, we denote by
\n\\[
\n\\Omega^p(M)=\\Gamma(\\Lambda^p(T^\\ast M))
\n\\]
\nbe the space of smooth $p$-forms over $M$.
\n\u00a0
\nBetween the $\\Omega^\\cdot(M,\\C)$, we have an operator $\\rm d$, i.e., the exterior differential operator, which maps a $p$-form to a $p+1$-form and with the property $\\rm d^2=0$. Then, we obtain a complex chain $(\\Omega^\\cdot(M,\\C),\\rd)$
\n\"\"<\/a>
\nDefine the $p$-th (complex coefficient) \\emph{de Rham cohomology} of $M$ as
\n\\[
\nH^p_{dR}(M,\\C)=\\frac{\\ker\\rd|_{\\Omega^p(M,\\C)}}
\n{\\rd(\\Omega^{p-1}(M,\\C))},
\n\\]
\nand the \\emph{total de Rham cohomology} of $M$ as
\n\\[
\nH^\\ast_{dR}(M;\\C)=\\bigoplus_{p=0}^{\\dim M}H_{dR}^p(M;\\C).
\n\\]
\nParticularly, a form $\\omega$ is called \\emph{closed} if $\\rd\\omega=0$; called \\emph{exact} if there exists an $\\eta$ such that $\\rd\\eta=\\omega$.
\n\\begin{excs}
\n For general $p$ the $p$-th de Rham cohomology may be hard to compute, but prove the following two special cases
\n \\begin{itemize}
\n \\item $H_{dR}^0(M;\\C)$ can be written as the direct sum of $\\C$ with multiplicity be its connected components, i.e.,
\n \\[
\n H_{dR}^0(M;\\C)=\\bigoplus_{\\textrm{connected components}}\\C.
\n \\]
\n \\item Suppose that $M$ be connected, and define an functional
\n \\[
\n \\int_M\\mathpunct{:}\\Omega^n(M;\\C)\\to\\C,
\n \\]
\n by $\\omega\\to\\int_M\\omega$, then prove that
\n \\[
\n \\ker\\int_M=\\rd(\\Omega^{n-1}(M;\\C)),
\n \\]
\n from which we can compute the $n$-th de Rham cohomology $H_{dR}^n(M;\\C)$. (cf. \\cite{bott1982differential})
\n \\end{itemize}
\n\\end{excs}
\nApparently $H_{dR}^\\ast(M;\\C)$ is a vector space, since for any $\\omega$, $\\omega’\\in\\Omega^\\ast(M;\\C)$ (maybe one is $k$ form and other is $k’$ form), then we can verify the following equation holds
\n\\[
\n[a\\omega]=a[\\omega],\\quad[\\omega+\\omega’]=[\\omega]+[\\omega’],
\n\\]
\nwhere $a$ is a constant function on $M$. With a litter more effort, we can prove that $H_{dR}^\\ast(M;\\C)$ is a ring. In fact, define
\n\\[
\n[\\omega]\\wedge[\\omega’]=[\\omega\\wedge\\omega’],
\n\\]
\nthen for any two differential forms $\\eta,\\eta’$ on $M$, we have (note that $\\rd\\omega=0=\\rd\\omega’$)
\n\\begin{align*}
\n(\\omega+\\rd\\eta)\\wedge(\\omega’+\\rd\\eta’)
\n&=\\omega\\wedge\\omega’+\\omega\\wedge\\rd\\eta’
\n+\\rd\\eta\\wedge\\omega’+\\rd\\eta\\wedge\\rd\\eta’\\\\
\n&=\\omega\\wedge\\omega’+\\rd((-1)^{|\\omega|}\\omega\\wedge\\eta’
\n+\\eta\\wedge\\omega’+\\eta\\wedge\\rd\\eta’),
\n\\end{align*}
\nwhich means
\n\\[
\n[\\omega]\\wedge[\\omega’]=[\\omega\\wedge\\omega’].
\n\\]
\nMoreover, $H^\\ast_{dR}(M;\\C)$ is \\emph{superexchange}, i.e.,
\n\\[
\n[\\omega]\\wedge[\\eta]
\n=(-1)^{|\\omega|\\cdot|\\eta|}[\\eta]\\wedge[\\omega].
\n\\]<\/p>\n

When $\\dim M=n=4m$, consider $H_{dR}^{2m}(M;\\C)$, then for any $[\\omega]$, $[\\eta]\\in H_{dR}^{2m}(M;\\C)$ we have
\n\\[
\n[\\omega]\\wedge[\\eta]=[\\eta]\\wedge[\\omega].
\n\\]
\nDefine a bi-linear operator $< \\cdot,\\cdot>$ on $H_{dR}^{2m}(M;\\C)$ as
\n\\[
\n< [\\omega],[\\eta]>=\\int_M\\omega\\wedge\\eta.
\n\\]
\nwe can prove that $< \\cdot,\\cdot>$ is a non-singular quadric on $H_{dR}^{2m}(M;\\C)$ and its signature is a topological invariants, called the \\emph{characteristic number} of $M$.<\/p>\n

Now, we state the \\emph{de Rham theory} without proof (for the proof cf. \\cite{bott1982differential}).
\n\\begin{thm}
\n If $M$ is a smooth closed orientable manifold, then for any integer $p$ with $0\\leq p\\leq\\dim M$, we have
\n \\begin{enumerate}
\n \\item $\\dim H_{dR}^p(M;\\C)<+\\infty$;\n \\item $H_{dR}^p(M;\\C)$ is canonically isomorphic to $H^p_{sing}(M;\\C)$, the $p$-th singular cohomology of $M$.\n \\end{enumerate}\n\\end{thm}\n\\begin{rem}\n The above Theorem shows that \\emph{algebraic topology} can be represented by \\emph{differential forms}, and \\emph{Chern-Weil} theory claims that the characteristic classes of vector bundle (which is a cohomology class) can be represented by \\emph{de Rham cohomology class}.\n\\end{rem}\n<\/p>\n","protected":false},"excerpt":{"rendered":"

Let $M$ be a smooth closed (i.e., compac …<\/p>\n

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