to make from the given object will be topologically equivalent.

![Topological equivalence of a coffee mug and a doughnut: A coffee mug can be [*smoothly*]{} deformed into a doughnut without tearing apart or joining two regions together.[]{data-label="fig:mug"}](mug.jpg)

A coffee mug is topologically equivalent to a doughnut, see FIG. \[fig:mug\]. Similarly, a football is topologically equivalent to the page of the book you are probably reading this article in. Some of you might have encountered the Gauss-Bonnet’s theorem in mathematics, $$\begin{aligned}
\label{gbth}
\frac{1}{4\pi}\int_M \kappa dA & = & 1-g\end{aligned}$$ Here $\kappa$ is the Gauss curvature of the object, $M$, we are considering. What this statement means is that if we integrate an object’s Gaussian curvature over the object’s surface, divide it by $4\pi$, the result will be $(1-g)$ where $g$ is an [*integer*]{} representing the number of holes in the object (or a manifold) $M$. Making small deformations on the surface of the manifold will of course change the curvature of the object [*locally*]{} but when we integrate the ‘changed’ curvature onto the deformed surface the right side of the equation still remains the same! Therefore, the number $g$ is known as a ‘topological invariant’ meaning it does not change by 