 Algebraic topology analyzes the holes and obstructions in connected topological spaces. A topologist is someone who imagines all objects to be made of unbreakable but very pliable playdough, and therefore does not see the need to distinguish between a coffee cup and doughnut because either can be turned into the other. Topology is concerned with the similarities and differences between coffee cups with \(n\) handles, surfaces with \(n\) holes, and more complicated shapes. Algebraic topology expresses the invariants of such shapes in terms of their homotopy groups and homology groups.