4 ms·

Yea, my own mental picture of homology and cohomology tends to default to DeRham (co)homology or the axiomatic approach versus the combinatorial approaches that

Yea, my own mental picture of homology and cohomology tends to default to DeRham (co)homology or the axiomatic approach versus the combinatorial approaches that are normally taught (I'm also bad at combinatorics). I'm going to be hand wavey here, but in DeRham, the "boundary" operator is basically just the derivative. So the DeRham homology of a manifold (the ker/im) are the functions that differentiate to zero (locally constant functions) / functions that can be integrated (eg functions that are the derivative of some function). You get left the functions that are "interesting" and define the geometry of the manifold in some way. Since manifolds are patched together pieces of Euclidean space, locally constant functions don't need to be globally constant which is why the quotient operation isn't trivially zero. If you've never seen quotient rings/groups/topologies before though, I can see how ker/im would be a major trip-up no matter the homology domain. The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. The DeRham homology and the simplicial homology give you the same homology group despite not having much to do with each other at first glance. (Stoke's theorem provides the isomorphism).

prionassembly 6y ago
Wikipedia is wonderful at giving you the handwavey picture, and then you can even skim some more advanced papers with handwavey ideas (e.g. there's a rigorous theory of PDEs in graphs relating the Khirchoff/incidence matrix to the de Rham operator, and then developing a theory of how graph Laplacians are really the Laplacians from calculus). But I kind of have a BDSM relationship with mathematics. I'm not happy until it makes me suffer.
auntienomen 6y ago
Wait until you learn about modern homotopy theory. It turns out that basically every cohomology theory can be described as homotopy classes of maps into some space. The standard cohomology groups with integer coefficients H^n(X; Z), those are just homotopy classes of maps from X into the Eilenberg-MacLane K(Z,n). Want K-theory instead? No problem. That cohomology group K(X) is homotopy classes of maps [X, Z x BU], where Z is the integers and BU is the classifying space for the unitary group. It turns out essentially all cohomology theories are represented this way. h(X) = [X,S_h], where [] is homotopy classes of maps and S_h is a topological space called the spectrum of the cohomology theory. Even wackier, the various algebra operations you can do on cohomology classes are mirrored by / derived from algebraic operations on the spectra. Which means that much of the machinery of algebraic geometry can be used to study cohomology theories.
rsj_hn 6y ago
> The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. This should not be crazy at all. The general recipe is that you take something for which there is a local solution, say in a ball or square, and then ask what are the obstructions that prevent you from patching the local solutions into a global solution. That is all that's going on with the ker/Im. For example, differential equations have local solutions (just the contraction mapping theorem). But can you patch them into global solutions? Locally spaces can be described as a ball or disk, but can you globally patch them together? So the obstructions end up measuring something topological. A series of local solutions on, say, a circle can patch together smoothly but when you wrap around to where you started you can be off - there is an obstruction. So the technique of finding local solutions can involve solving a differential equation or constructing a simplex but the obstructions of both have only to do with the consistency of patching these together, which is a topological question. As a historical note, mathematicians understood that counting the dimension of these obstructions reveals something about the underlying space, and there was hope that they would end up measuring finer structures -- e.g. the differential structure rather than just the topological structure. But the above discussion should at least motivate that this is unlikely. All the various *-homology theories ended up being mostly equivalent. It wasn't until the 1980s with the discovery of Simon Donaldson that true differential invariants were found, and the mechanism of finding these was to take the old homology invariants but apply them to an associated object whose own topology is shaped by the differential structure of the original space. For example, the moduli space of vector bundles associated to a space. Then the homological invariants of that space can detect differential invariants of the original space. There is a beautiful lecture series here: https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/ https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/

prionassembly 6y ago

Wikipedia is wonderful at giving you the handwavey picture, and then you can even skim some more advanced papers with handwavey ideas (e.g. there's a rigorous theory of PDEs in graphs relating the Khirchoff/incidence matrix to the de Rham operator, and then developing a theory of how graph Laplacians are really the Laplacians from calculus). But I kind of have a BDSM relationship with mathematics. I'm not happy until it makes me suffer.

auntienomen 6y ago

Wait until you learn about modern homotopy theory. It turns out that basically every cohomology theory can be described as homotopy classes of maps into some space. The standard cohomology groups with integer coefficients H^n(X; Z), those are just homotopy classes of maps from X into the Eilenberg-MacLane K(Z,n). Want K-theory instead? No problem. That cohomology group K(X) is homotopy classes of maps [X, Z x BU], where Z is the integers and BU is the classifying space for the unitary group. It turns out essentially all cohomology theories are represented this way. h(X) = [X,S_h], where [] is homotopy classes of maps and S_h is a topological space called the spectrum of the cohomology theory. Even wackier, the various algebra operations you can do on cohomology classes are mirrored by / derived from algebraic operations on the spectra. Which means that much of the machinery of algebraic geometry can be used to study cohomology theories.

rsj_hn 6y ago

> The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. This should not be crazy at all. The general recipe is that you take something for which there is a local solution, say in a ball or square, and then ask what are the obstructions that prevent you from patching the local solutions into a global solution. That is all that's going on with the ker/Im. For example, differential equations have local solutions (just the contraction mapping theorem). But can you patch them into global solutions? Locally spaces can be described as a ball or disk, but can you globally patch them together? So the obstructions end up measuring something topological. A series of local solutions on, say, a circle can patch together smoothly but when you wrap around to where you started you can be off - there is an obstruction. So the technique of finding local solutions can involve solving a differential equation or constructing a simplex but the obstructions of both have only to do with the consistency of patching these together, which is a topological question. As a historical note, mathematicians understood that counting the dimension of these obstructions reveals something about the underlying space, and there was hope that they would end up measuring finer structures -- e.g. the differential structure rather than just the topological structure. But the above discussion should at least motivate that this is unlikely. All the various *-homology theories ended up being mostly equivalent. It wasn't until the 1980s with the discovery of Simon Donaldson that true differential invariants were found, and the mechanism of finding these was to take the old homology invariants but apply them to an associated object whose own topology is shaped by the differential structure of the original space. For example, the moduli space of vector bundles associated to a space. Then the homological invariants of that space can detect differential invariants of the original space. There is a beautiful lecture series here: https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/ https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/