Differentiation between Geometry and Topology in Mathematics

About the Study

Geometry and topology is a general term in mathematics that refers to the historically separate fields of geometry and topology. This is because general frameworks allow both fields to be uniformly manipulated, most notably in the local to global theorems in Riemannian geometry and in discoveries like the Gauss-Bonnet theorem and Chern-Weil theory.

It differs from "geometric topology," which is a more limited definition of topological applications to geometry.

It contains:

•Differential topology and geometry

•Mathematical topology (including low-dimensional topology and surgery theory).

Although some sections of geometry and topology (such as surgery theory, particularly algebraic surgery theory) are extensively algebraic, it does not contain aspects of algebraic topology like homotopy theory.

Difference between topology and geometry

Topology only has global structure, but geometry also contains local (or infinitesimal) structure. Alternatively, topology has discrete moduli whereas geometry has continuous moduli.

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

Symplectic manifolds are a boundary case, and coarse geometry is global, not local, therefore the terminology is not entirely coherent.

Moduli

A structure is considered to be rigid if it possesses discrete moduli, has no deformations, or has deformations that are isomorphic to the original structure. Topology is the study of rigid structures, whether they are geometric or topological. The structure is considered flexible and the study of it is geometry if it exhibits non-trivial deformations. Studying maps up to homotopy involves topology because the set of homotopy classes of maps is discontinuous. Similarly, the moduli of differentiable structures on exotic R4s are continuous as opposed to the discrete spaces that characterize differentiable structures on a manifold, which are examples of topology.

Since algebraic varieties have continuous moduli spaces, algebraic geometry is the study of them. These moduli spaces have limited dimensions. An infinite-dimensional space characterizes the set of Riemannian metrics on a certain differentiable manifold.

Symplectic manifolds

As a boundary case, symplectic manifolds are the subject of the fields of symplectic topology and symplectic geometry. A symplectic manifold has no local structure according to Darboux's theorem, which argues that their field of study should be named topology.

The space of symplectic structures on a manifold, in contrast, forms continuous moduli, suggesting that the study of these structures should be referred to as geometry. Therefore, the space of symplectic structures is discrete up to isotopy.

In high and low dimension, manifolds interact in drastically different ways. Manifolds of dimension 5 and above, or embeddings in codimension 3 and up, are referred to as having a high-dimensional topology. Questions in dimensions up to 4, or embeddings in dimensions up to 2, are the focus of low-dimensional topology. Because it is both high-dimensional (topologically) and low-dimensional (differentially), dimension 4 is unique in that, it produces phenomena that are unique to that dimension, such as unusual differentiable structures on R⁴.

Important tools in geometric topology

The fundamental group of a manifold determines much of the structure and is a crucial invariant in all dimensions; in dimensions 1, 2, and 3, the range of possible fundamental groups is constrained, while in dimensions 4 and higher, every finitely presented group is the fundamental group of a manifold note that it is sufficient to show this for 4 and 5 dimensional manifolds, and then to take products with spheres to get higher ones.

A linked orientable manifold has precisely two different alternative orientations. A manifold is orientable if it has a consistent option of orientation. Depending on the desired application and level of generality, many comparable formulations of orientability might be presented in this context. For generic topological manifolds, homology theory-based formulations are frequently used; however, for differentiable manifolds, where more structure is available, a formulation in terms of differential forms is possible. The idea of orientability of a family of spaces that are parameterized by another space (a fibre bundle) and for which an orientation must be chosen in each of the spaces that continuously varies with respect to changes in the parameter values is a key generalisation of the concept of orientability of a space.