Thread by @pursuingstacks

A thread on de Rham's Theorem from Differential Topology:

"Euclid missed a great opportunity here: if he had stated the principle , 'The extremity of an extremity is empty' ,he could be considered as the discoverer of the BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d•d = 0." 1/n

This basic equation of Homological Algebra is just an abstract incarnation of something we encounter at different places in Mathematics. Like, in Topology given a space X with boundary, one can not construct boundary of that boundary 2/n

( imagine a Unit disc for example, its boundary is a circle but circle itself has no boundary ),next in Vector Calculus,we all learn things like div(curl)F=0 and curl(grad)f=0,where F and f are vector/scalar fields on a X (with some extra structure to do Calculus on it) 3/n

One might wonder if the previous two incarnations of the same equation d•d=0 are in some sense related? Afterall the underlying space involved in both the cases is X (with some extra structure ). As it turns out, there's a positive answer to that. 4/n

Before i state the actual correspondence lets try to appreciate what this correspondence would eventually achieve. In Topology, we are very Interested in figuring out when two different looking spaces are same upto a given condition/property? 5/n

and further we would like to Classify spaces upto that property. This is an extremely difficult task unless we make it simple by attaching to various spaces, objects which we can handle easily. More precisely these objects are Algebraic in nature called (Co)homology groups. 6/n

So now we can compute (Co)homology groups of two different spaces and see if we get the same answer and if we do get the same answer then underlying spaces share some Properties. In particular, let two spaces be a punctured disc (disc with one point missing ) and circle 7/n

at first these two spaces look very different but their (Co)homology groups are same and that is Z (group of Integers ), what this tells us is that both spaces have 1 hole. 8/n

Further, the way we actually compute these groups involves the equation d•d=0. Now recall I mentioned two different incarnations of this equation, that means we get two ways of extracting information about X - argument ). 9/n

one by considering space X itself ( that boundary of boundary Argument ) and the other by considering the extra structure on X which allows us to do Calculus on it (Vector Calculus Argument ) 10/n

These two ways give us two different (Co)homology groups. The first incarnation leads to what we call Singular (co)homology groups and the other one leads to de Rham (co)homology groups. Now let's state that beautiful correspondence we are after. 11/n

de Rham's Theorem-

Singular (Co)homology groups are isomorphic( same for all purposes) to de Rham (Co)homology groups.

Moral of the Story:
Calculus on space X somehow gives us useful information about the Topology of X. 12/n , n=12