Topology- From Euler to Poincaré

Turning a sphere inside out is one easy task. Simply poke a hole in it and yank it through. Or is it? Can it be done without poking a hole? And with a material that can bend, stretch and pass through itself? Enter topology; where doughnuts meet coffee mugs and where pretzels become nontrivial trefoils. It's way of looking at space with a sense of 'sameness' and connectivity, where even gravity is reduced to curvature on a trampoline. Indeed topology is the product of the cross fertilization of graph and knot theory, in 1736 Euler established the impossibility of the circuit round the seven bridges of Königsberg; by envisaging each plot of land as a vertex connected by lines, it became clear that an even number of edges was needed to cross each bridge once and fulfill the problem. But since the vertices all had odd numbers of edges, it was out of the question. More imaginary was the discovery of the Möbius strip, a surface with a single edge and a single side. Such a surface emerges via twisting a strip so that one side is reversed and formed into a closed loop, this introduced the notion of a non-orientable plane that confuses the sense of 'inside' and 'outside'. Attaching two Möbius strips together results in a Klein bottle, a closed surface lacking a defined boundary and may also be formed abstractly by gluing the remaining edges of a cylinder in opposite directions to make a torus. But to achieve this is everyday Euclidean space, the surface would have to go through itself to situate the edges. More intriguing was the Poincaré conjecture posited in 1904 and the topology of the sphere; simply put, the simplest object which is closed in any number of given dimensions. Any closed 3D manifold where a loop may be contracted to a single point is a 3D sphere, whereas loops on a torus fail to contract to a single point. But the central issue at large remained whether a closed 3D manifold could exist independent of the sphere but by considering Thurston's geometrisation theorem, where 8 types of manifolds could be sewn to form other 3D surfaces and Ricci flow, whereby irregular and lopsided spaces can be turned into uniform ones, the concept of a non-spherical closed manifold could be ruled out. Algebraic topology also introduces the idea of a vector bundle, considering spaces and structures described above a surface as opposed to within it; by designating a vector space to each point, followed by a fibre, a vector field is produced. Suppose a series of vectors lying tangent on a sphere, at least a single point will remain with a vector value of zero is inevitable making a hairy torus easily combable whereas a hairy ball will produce a crown at each pole. Moreover, at even the simplest topologies; the concept of homeomorphism forms a continuous transformation function between the torus and the coffee mug, both objects are said to be homotopic just as the 2D sphere and the horned sphere can be converted into one another without tearing or incising. Homotopy may also be applied when fine spaces or holes cannot be accounted by via homology or by examining features of the topological space that lack a boundary and are not boundaries in themselves.