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Suppose $X$ is a topological space and I want to talk about its “homology”.

There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute.

Suppose we assume that there is an extra structure on the topological space $X$, namely the structure of a CW complex, then, we can talk about the notion of cellular chain complex and from there the notion of cellular homology. This is easier than computing singular homology.

Suppose further that this topological space $X$ (that we assumed to have CW structure) has an extra structure of a simplicial complex, then, we can talk about the notion of simplicial chain complex and then the notion of simplicial homology. This is easier to compute than cellular homology.

Then it is standard result that any two homology groups coming from different approaches coincide when both of them makes sense.

Question : Is there an extra structure (non trivial) I can add on a space having simplicial structure that makes it easier to compute homology in terms of chain complex simpler than simplicial chain complex?

Same is the situation with cohomology. Suppose I have a topological space $X$, I can talk about its singular cochain complex and the corresponding singular cohomology.

Suppose that this topological space $X$ has structure of a manifold, then, we can talk about the cochain complex of differential forms and use it to compute the cohomology of the topological space $X$. It is a standard result that, with correct coefficients, singular cohomology is same as deRham cohomology (deRham’s theorem).

Question : Is there an extra structure that I can add on a manifold that gives a simpler cochain complex than cochain complex of differential forms that gives an easier way to compute cohomology of the manifold? For example, suppose I fix a connection on the tangent bundle $TM\rightarrow M$ of the manifold $M$ (or a Riemannian metric on the manifold $M$), can I produce a simpler cochain complex using the connection that computes cohomology easily?

If I am trying to make sense of notion of cohomology theory fixing a connection on the tangent bundle $TM\rightarrow M$ (a metric on the manifold $M$) then it is reasonable to expect that this notion should not be dependent on the choice of connection I have fixed. Does assuming that there exists a flat connection on the tangent bundle suggest some obvious cochain complex?

Suppose I ask that the manifold $M$ has an extra structure of a Lie group then there is a simpler cochain complex that computes the cohomology of the manifold easier than deRham cohomology. This is too much to ask, I am looking for results that comes in between deRham cohomology of manifold and strictly lesser structure than the notion of Lie groups.

Any references are welcome.

Edit : May be it is clear from the way I have written but by "an extra structure on manifold $M$", I do not mean for example, an action of a Lie group $G$ on the manifold $M$, that gives the equivariant cohomology $H^k((EG\times M)/G)$.

Edit : I am really interested in knowing the answer for what I asked above. Any pointers to references are also welcome.

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  • $\begingroup$ On a manifold, I might have a Morse function procuding the Morse chain complex. On a simplicial complex I might have a discrete Morse function, which can drastically simplify the simplicial chain complex. $\endgroup$ – Lennart Meier Oct 14 '19 at 6:27
  • $\begingroup$ @LennartMeier I do not know much about Morse function.. Is it reasonable to ask a manifold has a Morse function, I mean does it happens quite often that a manifold I choose has a Morse function.. please see if you can add it as an answer.. can you suggest some references, I am already seeing Wikipedia page.. $\endgroup$ – Praphulla Koushik Oct 14 '19 at 6:35
  • $\begingroup$ Please let me know reason for downvote.. what is the use of that downvote if it does not reach me anything $\endgroup$ – Praphulla Koushik Oct 15 '19 at 2:32
  • $\begingroup$ Read reach as teach $\endgroup$ – Praphulla Koushik Oct 15 '19 at 3:24
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    $\begingroup$ Often times computing homology via a CW-structure, triangulation or a Morse function are equally difficult. For example, with triangulations your attaching maps are simpler, but you have far more of them than with a CW-complex. Is one "easier" than the other? This is analogous to the trade-off between runtime and memory usage in software optimization. A slower algorithm that uses less memory is best if you have little memory, but a fast algorithm that's a memory hog could be better if you have access to unlimited memory. $\endgroup$ – Ryan Budney Oct 17 '19 at 5:46
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Given a Morse function on a Riemannian smooth manifold, you obtain (under mild conditions) a chain complex computing its homology, the Morse complex. Its generators are the critical points and differentials are determined via flow lines. Morse functions always exists (see e.g. the classic book by Milnor on Morse theory). Milnor's book does not treat Morse homology though. You find some references on the wikipedia page on Morse homology. Another source is the book Morse homology by Schwarz.

On the other hand you might be given a simplicial complex. This already provides you with a chain complex computing its homology, but this might be very big. Using ideas from Morse theory, Forman developed discrete Morse theory. (Although ideas like this have been floating around for a longer time, e.g. by K.S. Brown.) A source is for example: A user's guide to discrete Morse theory Discrete Morse theory can dramatically simplify the chain complex for the computation of simplicial homology. This has been used a lot for computations with persistent homology recently. (Vidit Nanda is a name coming to my mind who is working in this direction.)

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  • $\begingroup$ Thanks for your answer. I will try to read definitions and statements in Minor's book and try to get something... $\endgroup$ – Praphulla Koushik Oct 14 '19 at 7:57

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