I’m pretty interested in the history of mathematics, and it has always been my belief that the great pioneers of Differential Geometry were Gauss and Riemann, and the father of topology was mostly accredited to Poincare. However, now I am taking a differential topology/geometry sequence and we basically learn two semesters of Differential Topology as a prerequisite for Differential Geometry, and this confuses me since Riemann and Gauss were both dead before Poincare could’ve done anything with topology. So was Differential Geometry created before Differential Topology? If so how were Gauss and Riemann able to do it before what seems to be the necessary definitions of topological spaces, and smooth manifolds?
Also if anyone has any sources on a history of modern mathematics (like 18-21st century) that they could recommend, that would be very cool since I really can’t seem to find any that really get into the weeds as much as I would like.