A Manifold is an n-dimensional topological space which is locally Euclidean; meaning that the vicinity around each point resembles Euclidean space. So for an n-dimensional manifold, the space around each point is topologically equivalent to the Euclidean space of dimension n. Thus, one of the main objectives of studying manifolds is to find different methods to distinguish manifolds from each other. Although such a feat is much easier to do in higher dimensions, it becomes increasingly difficult in low dimensional topology – the realm of shapes and forms which we are familiar with.
To illustrate this idea, consider two different manifolds, a circle and a closed loop. Although they are different manifolds, they are topologically the same. However, a ball and a closed loop, such as a donut are topologically different. One way to underscore the differences or similarities is to implement the Loop Test. No matter where the loop is tied around the ball, the tighter and tighter the loop gets, it will eventually fall off the ball. This phenomenon does not occur for a donut, whose surface is called a “torus.” No matter how tight the loop gets, the string will not fall off the donut unless the shape/surface of it is disrupted or cut. Thus, it can be seen that these two entities are different topologically.
Apart from dimensional classification, manifolds can also be distinguished through many other properties such as whether it is compact or non-compact, connected or disconnected, smooth or complex. Thus, understanding Manifolds is crucial for the study of geometric and topological analyses.