Geometric topology

Geometric topology studies manifolds, their embeddings and the ways they can be assembled, classified or transformed. Typical questions ask whether two manifolds are homeomorphic or diffeomorphic, whether two embeddings are related by an ambient isotopy, and how gluing simpler pieces changes the resulting space. The category of structure matters: a homeomorphism preserves topology, while a diffeomorphism also preserves smooth structure. Algebraic invariants and geometric constructions work together in these questions.^[ref]

For example, a closed connected oriented surface is classified by its genus of a surface $g$, with Euler characteristic $\chi =2-2g$. A sphere, torus and genus-two surface have Euler characteristics $2,0,-2$. A polygon construction for the genus-two surface has one vertex, four edges and one face, giving $1-4+1=-2$. Specifying closed, connected and oriented is essential; surfaces with boundary or noncompact surfaces require more data.^[ref]

In 1956 John Milnor constructed smooth seven-dimensional manifolds homeomorphic to the standard seven-sphere but not diffeomorphic to it. His construction used three-sphere bundles over the four-sphere and an invariant distinguishing smooth structures.^[ref] This shows why classifying the underlying topological space and classifying its smooth structures are distinct problems.