Daily Differential Geometry

Tomorrow's Problem.
Please vote to select what problem is solved tomorrow.
	Problem(18-7) The Poincaré Duality Theorem: Let $M$ be an oriented smooth n-manifold. Define a map $\mathrm{PD}: \Omega^p(M) \to \Omega_c^{n-p}(M)^$ by \[ \mathrm{PD}(\omega)(\eta) = \int_M \omega \wedge \eta. \] The map $\mathrm{PD}$ descends to a linear map $\mathrm{PD}: H^p_{\Omega}(M) \to H^{n-p}_c(M)^$. Moreover, the induced map $\mathrm{PD}$ is an isomorphism for each $p$.
	Problem(18-8) Let $M$ be a compact smooth $n$-manifold. The de Rham groups $H^p_\Omega(M)$ are finite-dimensional for every $p$, and if $M$ is orientable, then $\Dimension H^p_\Omega(M) = \Dimension H^{n-p}_\Omega(M)$.
	Problem(18-9) Let $M$ be a smooth $n$-manifold, such that all the de Rham groups $H^p_\Omega(M)$ are finite-dimensional. The Euler characteristic of $M$ is the number \[ \chi(M) = \sum_{p = 0}^n (-1)^p \Dimension H^p_\Omega(M). \] Then $\chi(M)$ is a homotopy invariant of $M$, and $\chi(M) = 0$ when $M$ is compact, orientable, and odd-dimensional.

Today’s Solution.
John M. Lee — Introduction to Smooth Manifolds — Problem(11-17)
Let $\mathbb{T}^n = \mathbb{S}^1 \times \cdots \mathbb{S}^1 \subset \mathbb{C}^n$ denote the $n$-torus. For each $j = 1, \dots, n$, let $\gamma_j: [0,1] \to \mathbb{T}^n$ be the smooth curve segment \[ \gamma_j: t \mapsto \left( 1,\ \dots,\ e^{2\pi i t},\ \dots,\ 1 \right), \] such that $e^{2\pi i t}$ is in the $j$th place. Suppose $\omega \in \mathfrak{X}^*(\mathbb{T}^n)$ is a smooth covector field on $\mathbb{T}^n$ that is closed, meaning the component functions of $\omega$ satisfy \[ \frac{\partial \omega_i}{\partial x^j} = \frac{\partial \omega_j}{\partial x^i} \] for each pair of indices $i$ and $j$. Then $\omega$ is exact if and only if $\int_{\gamma_j} \omega$ vanishes for all $j$.
Solution