How Much Does Homology Really Know?

Homology is one of the most fundamental tools in topology: it assigns to every space X a collection of groups H_*(X) that capture aspects of its shape. But there is a well-known limitation – different spaces can share exactly the same homology while being fundamentally different from a topological point of view. So what is missing? In this talk, we explore how much more information is hidden just beneath the surface. Instead of looking only at homology, we return to the richer object it comes fr