Geometric Topology Vs. Algebraic Topology.

[MathematicalPhysicist]

i know that geometric topology is a field that is connected to knot theory, i wonder what are the similarities between the two subjects, and in what subject in particular they overlap?

 

[matt grime]

It would appear that the objects of study of geometric topology are a subset of algebraic topology. That subset consisting of low dimensional manifolds. There is presumably, therefore, more of an analytic flavour to its study than general algebraic topology.

 

[tacman]

Geometric topology and algebraic topology are two closely related fields within the broader subject of topology. Both fields study the properties and structure of topological spaces, such as surfaces and higher-dimensional manifolds. However, they approach these objects from different perspectives and use different tools and techniques.

Geometric topology is primarily concerned with the study of the shape and structure of topological spaces. It focuses on the properties of these spaces that are preserved under continuous deformations, such as stretching, bending, and twisting. This field includes subfields such as knot theory, which studies the properties of mathematical knots and their embeddings in three-dimensional space.

On the other hand, algebraic topology uses algebraic techniques to study topological spaces. It assigns algebraic invariants, such as homology and cohomology groups, to topological spaces and uses these invariants to distinguish between different spaces. This allows for a more abstract and algebraic approach to understanding topological spaces and their properties.

Despite their differences, geometric topology and algebraic topology have significant overlap in their subject matter. For example, knot theory is a subfield of geometric topology that also has connections to algebraic topology. In particular, knot invariants, such as the Alexander polynomial and the Jones polynomial, can be computed using techniques from algebraic topology.

Similarly, the study of surfaces and higher-dimensional manifolds in geometric topology also has connections to algebraic topology. For instance, the fundamental group, a key topological invariant, can be computed using algebraic methods.

Overall, while geometric topology and algebraic topology have distinct approaches, they both contribute to our understanding of topological spaces and have significant overlap in their subject matter.