Abstract

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 from: the singular chains of a space. These chains carry additional structure that records how pieces of the space interact with one another – structure that disappears when passing to homology, but is responsible for subtle phenomena familiar to topologists.

The surprising message is that this extra structure is powerful enough to recover the entire space. More precisely, when singular cochains are viewed with all of their higher-order structure (as an E_\infty-coalgebra), they form a complete invariant: from this data alone, one can reconstruct the homotopy type of the space in a functorial way.

This result builds on and extends ideas from rational homotopy theory and deep work of Mandell, Yuan, and Bachmann–Hahn. At its core lies a classification of particularly well-behaved (“perfect”) E_\infty-coalgebras, which makes this reconstruction possible.

This is joint work with F. Riedel.

A wine reception in the Central Core will follow the lecture.