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jwiltshiregordon.github.io

5 pointsjdwg9 comments

Another basic Q. (sorry, I am comfortable with the Euler characteristic, and with filling holes in STL models, but not with homology):

Taking chains to be a rough analogue of functions, could we say that cycles on (X,A) serve as a kind of dual space for A? (because if we ignore intermediates, they're basically connecting A<->A?) So from the dual side, we seem to have the case that an appropriate quotient might be A/X?

(maybe an alternate view of what I'm trying to ask about: in Mathematics Made Difficult, we find many proofs of [something in A implies something else in A] which, instead of being limited in scope to A and its subfields as is customary, wander leisurely through X to ultimately prove the inclusion in A)

Cycles on (X, A) really can connect parts of A to other parts that look remote if you are forced to stay within A. This is visible in the long exact sequence; when the third map is nonzero, you have cycles in A that become boundaries when you allow chains in X.

Sorry, I do not see what you mean by "dual space," and I do not myself view chains as a analogous to functions.

Thanks for drawing my attention to the long exact sequence*; if I'm interpreting it correctly, the injection at 𝛼 and projection at 𝛽 split, so we have a direct sum: H(X,∅) ≃ H(X,A) ⊕ H(A,∅)?

(never mind the function analogy, I was trying to handwave a quotient in the other direction but that would fall immediately out of the direct sum if I understand correctly: A ≃ (A⊕B)/B and B ≃ (A⊕B)/A?)

* which forms a Möbius band in its own way, because at H(A,∅) we feed it into 𝛼 as ∅ ⊕ Cycles(A) but get it out of 𝛾 as Cycles(A) ⊕ ∅, leading to a "twist"?

[Edit: are you aware of Dan Piponi's blogging?

http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell...

http://blog.sigfpe.com/2006/08/what-can-we-measure-part-i.ht...

http://blog.sigfpe.com/2010/01/target-enumeration-with-euler...

etc.]

The direct sum decomposition you mention doesn't always happen, as you can see in some of the examples in the calculator. It is closer to happening when \gamma is zero, and if \gamma is zero and we switch from integer coefficient to field coefficients, then it always happens!

obviously I need to play with the calculator more (and probably work the exercises) ... hope you don't mind if I get back to you with more Q's once I have them.

well, I love the subject, so I'm glad to discuss whatever! And let me know if all of the exercises are too hard. There should always be a couple of easy ones, but I don't know this audience very well.

> A cycle on a pair (X,A) is a chain on X whose boundary is a chain on A .

is this roughly saying that a cycle on (X,A) is a chain on X that would be a cycle on the quotient X/A?

Yes, this is true (or nearly) for simplicial complexes. A cycle on (X, A) is the same as a cycle on (X/A, A/A) = (X/A, single point), and this is a useful point of view, since X/A can be much simpler to visualize.

Interactive explanation of simplicial homology and lets you compute the long exact sequence of a pair

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