From (algebraic) topology to geometry I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an a...--prophetes.ai

From (algebraic) topology to geometry I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an abstraction of geometry. So how can we link back? For example, when we have a chain complex, how can we introduce formally an affine space and have this complex endowed with geometric features? Cheers & Thanks!

Chain complex happen to be the simplicial objects in the category of modules. To any simplicial object, it is possible to construct funtorially a topological space, its _geometric realization_.

This functor has the very nice property to be a left ajoint to the singular functor $S$ which associates to any topological space, its singular chain complex.

$$ \mathrm{Hom}_{\mathrm{Top}} (|X|, Y) = \mathrm{Hom}_{\mathrm{Ch}}(X, SY)$$