Geometry and topology

In

Sharp distinctions between geometry and topology can be drawn, however, as discussed below.^{[clarification needed]}

It is also the title of a journal Geometry & Topology that covers these topics.

Scope

It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry.

It includes:

• Differential geometry and topology
• Geometric topology (including low-dimensional topology and surgery theory)

It does not include such parts of

Distinction between geometry and topology

Geometry has local structure (or infinitesimal), while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli.

By examples, an example of geometry is Riemannian geometry, while an example of topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology.

The terms are not used completely consistently: symplectic manifolds are a boundary case, and coarse geometry is global, not local.

Local versus global structure

By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature.

By contrast, the curvature of a Riemannian manifold is a local (indeed, infinitesimal) invariant^{[clarification needed]} (and is the only local invariant under isometry).

Moduli

If a structure has a discrete moduli (if it has no deformations, or if a deformation of a structure is isomorphic to the original structure), the structure is said to be rigid, and its study (if it is a geometric or topological structure) is topology. If it has non-trivial deformations, the structure is said to be flexible, and its study is geometry.

The space of homotopy classes of maps is discrete,^[a] so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R⁴s have continuous moduli of differentiable structures.

By Darboux's theorem, a symplectic manifold has no local structure, which suggests that their study be called topology.

By contrast, the space of symplectic structures on a manifold form a continuous moduli, which suggests that their study be called geometry.

However, up to isotopy, the space of symplectic structures is discrete (any family of symplectic structures are isotopic).^[1]

Notes