= 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.<Topics in Geometric Topology by Jacob Lurie>[Lecture 1, classification of manifolds and spaces of diffeomorphisms]{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.<Topics in Geometric Topology by Jacob Lurie>[Lecture 1, opening classification example]{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.<On Manifolds Homeomorphic to the 7-Sphere by John Milnor>[Introduction and section 3, Theorem 3, printed page 403]{ref} This shows why classifying the underlying topological space and classifying its smooth structures are distinct problems.
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