Cohomology-Partition of Unity

Smooth manifolds are defined on nice topological spaces that have countable basis and are Hausdorff spaces. Partition of unity (POU) can be defined and existence proven more easily.

 Idea behind POU is as follows. Imagine a set of continuous functions defined on a topological space that takes values in the closed interval [0,1]. They are sort of functions that finitely many of them take non-zero values in a small neighborhood and vanish (become 0) in other places.

So how is all the above formalized? Given the niceness of topological space, we can defined these functions in the following manner.

1. Take an open cover $U_i$ of manifold $M$ such that at each point $x$ there
are finitely many open covers $U_i$.
2. Take a family of differential functions $0 \leq \beta_i(x)  \leq 1$ such that sum is unity at point x.
Such a family ${\beta_i(x)}$ define  on ${U_i}$ called  a POC subordinate to ${U_i}$.
3. These functions vanish beyond the ${U_i}$.
How's this useful?
On manifolds, integration of the forms can be done on a coordinate patch. POU helps extend this coordinate patch integration to whole manifold.
For an orientable manifold $M$, take a volume form $\omega$ at a point $p$.
$$\begin{equation}   \omega = h(p) dx^1 \wedge dx^2 \wedge \cdots dx^n \end{equation}$$
with postive definite $h(p)$ on a chart $U_i$ whose coordinate is $x=\phi(p)$.

Let $f:M \rightarrow R$ be a function on $M$. In the coordinate neighbourhood $U_i$ one can define integration of $n$-form as
$$\begin{equation}   \int_{U_i}f = \int_{\phi_i(U_{i})} f(\phi^{-1}_i(x))h(\phi^{-1}_i(x))dx^1 \cdots dx^n \end{equation}$$
POU enables us to extend this integration over entire manifold.

Example: On $S^1$ , use open covers $U_1=S^1 - (1,0)$ and $U_2=S^1-(-1,0)$ to define "bump" functions
$\beta_1(\theta)=sin^2(\frac{\theta}{2})$ and $\beta_2(\theta)=cos^2(\frac{\theta}{2})$.
Since values of $sin$ and $cos$ belong to interval $[0,1]$ and sum of $\beta_1,\beta_2$ is always unity, it is easy to see that this a POU.

We can use this to integrate - say $\int_S^1 cos^2\theta$ over $S^1$.

$\beta_1(\theta)=sin^2(\frac{\theta}{2})$ and $\beta_2(\theta)=cos^2(\frac{\theta}{2})$.
This is a partition of Unity because,
1. $\beta_i(\theta) \in [0,1]$ for $i=1,2$.
2. if $p=(1,0)$ as $p \notin U_1$ results in value of function $0$. Similarly,
for $U_2$, at $p=(-1,0)$, value is $0$.
3. At any angle where both the functions are defined, POU acts as a weight that allocates a fraction to one function and rest of the fraction to next function. In our example, say we take $\theta = \pi/4$, $\beta_i(\pi/4)=1/2$, hence sum of $1/2+1/2=1$.

 We can use this to integrate - say $\int_{S^1} cos^2\theta$. Let us see what happens. Since POU acts as a weighting function that sums to $1$, we can allow overlaps in the integration. POU will allocate appropriate weight and make sure that the integration works as expected.
$$\begin{equation}   \int_{s^1}cos^2(\theta)d\theta = \int_0^{2\pi}sin^2(\theta/2)cos^2(\theta)d\theta + \int_{-pi}^{pi} cos^2(\theta/2)cos^2(\theta)d\theta = \pi \end{equation}$$