Geometric Topology

Geometric topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous transformations, such as stretching and bending, but not tearing or gluing. It focuses on the shapes and structures of objects, like surfaces and manifolds, to understand their fundamental characteristics. For instance, it explores how a coffee cup can be transformed into a doughnut shape, since both have one hole. Geometric topology combines ideas from geometry and algebra, helping mathematicians classify spaces and solve complex problems about their shapes and dimensions.