3 Matching Annotations
1. Dec 2021
2. www.quantamagazine.org www.quantamagazine.org
1. “[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight,” said Agustin Moreno of the Institute for Advanced Study.

Floer homology theory depends only on the topology of the manifold.

#### URL

3. Jun 2021
4. math.stackexchange.com math.stackexchange.com
1. Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R

Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==

#### URL

5. en.wikipedia.org en.wikipedia.org
1. To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure.

Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.