/sci/ - Science & Math

>>11360543
For each [math]X\in \mathcal{H}(W)[/math], let [math]f_\alpha \in C^\infty(U_\alpha)[/math] be the associated Hamiltonian function on the local patch [math]U_\alpha \subset W[/math]. By definition we have [math](\iota_X\omega - df)_{U_\alpha} = 0[/math]; indeed, this is Poincare's lemma stating that, since the Lie derivative [math]L_X\omega = 0 = d\iota_X \omega + \iota_X d\omega = d\iota_X \omega[/math] and [math]\iota_X d\omega[/math] is closed, [math]\iota_X\omega \in C^1(W)[/math] is locally exact. Hence we know that [math]\operatorname{ker}(\mathcal{H}_\text{loc}(W)\rightarrow H^1(W)) \cong \mathcal{H}(W)[/math] is the space of [math]X[/math] with globally defined Hamiltonians; namely the [math]f_\alpha[/math]'s can be patched together.
What is the obstruction to this patching? It has to do with coordinate transitions [math]g_{\alpha\beta}[/math] on overlaps [math]U_\alpha\cap U_\beta[/math], or alternatively [math]g_{\alpha\beta} = f_\alpha^{-1} df_\beta[/math], which can be trivialized if the log function is well-defined [math]g=d\ln f[/math] for [math]f = f_\alpha|_{U_\alpha \cap U_\beta} = f_\beta|_{U_\alpha\cap U_\beta}[/math]. Of course, as we all know, classes of these obstructions is the first Cech cohomology [math]\check{H}^1(W)[/math], which over [math]\mathbb{R}[/math] is de Rham [math]\check{H}^1(W) \cong H_\text{dR}^1(W)[/math].
Alternatively one can notice that, since both Cech and de Rham satisfies the dimensionality axiom (in addition to the EM axioms) of cohomology theory, the fact that the coefficients lie in a field of characteristic 0 means that, on finite CW spaces, there is only one cohomology theory: the singular cohomology [math]H^1(W,\mathbb{R})[/math]; hence [math]\check{H}^1(W) \cong H^1(W) \cong H^1_\text{dR}[/math].