Reconstructing Topological Spaces by Algebraic Images of Functors.

Question

At the beginning paragraph of chapter 1 in Hatcher’s Algebraic Topology it says:

Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images — the ‘lanterns’ of algebraic topology, one might say — are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topo- logically related spaces have algebraically related images.

With suitably constructed lanterns one might hope to be able to form images with enough detail to reconstruct accurately the shapes of all spaces, or at least of large and interesting classes of spaces.This is one of the main goals of algebraic topology, and to a surprising extent this goal is achieved.

I attempted to search for more information about this method of reconstruction but failed to find anything. Could anyone please give some details of this reconstruction of topological spaces?

Answer 1

I am not an algebraic topologist, but to me it seems like the more accurate word would be 'identify' rather than 'reconstruct'. Assume you are interested in a certain class of 'shapes' (e.g. orientable closed surfaces) and you already have collection of basic examples of those shapes $X_1,X_2,\dots$ (e.g. a sphere, a torus, $\dots$). Now, if I hand you another of those shapes $Y$, can you identify it with one of these $X_i$? In the example at hand you might want to suggest to just count the holes of $Y$: If it has zero holes, then it is a sphere, if it has one hole, then it is a torus and so on. Counting holes can be formalized by one of the techniques that Hatcher refers to and indeed for this class of shapes (orientable closes surfaces) it is the only thing you need to successfully identify objects.

For other classes of shapes (e.g. higher dimensional manifolds, arbitrary toplogical spaces, knots, graphs, ...) you need more tools (lanterns) to identify objects. Unfortunately it is rarely possible to give an exhaustive list of all possible shapes (like the $X_1,X_2,\dots$), but often you can still use your lanterns to at least tell whether two given shapes $X$ and $Y$ can be the same. A typical argument is to say that if they appear differently under a certain lantern, they cannot be the same. (The question of whether you have enough lanterns to conclude that they are the same once you you have checked under each of it is much harder, though.)