Exact Sequences: Intuition & Theory

[StatusX]

This is a very vague question, but I'd like to know whatever insights anyone could offer about exact sequences. What do they represent? Why are they so important? I'm studying homology right now, and exact sequences are central to the theory, but I've never seen them before. What is the intuition behind the different lemmas (snake, five, etc), or are they just abstract diagram chasing with no easy interpretation?

 

[matt grime]

The point about homology is that it measures defects, or obstructions.

So suppose that I have a map of vector spaces, f:V-->W. What is the obstruction that prevents that being an isomorphism? Well, there is the kernel of f, stopping it being injective and the cokernel of f stopping it being surjective. What that means is that we can complete

V-->W

to an exact sequence

0-->ker(f)-->V-->W-->coker(f)-->0

Anyhow. getting back to things at hand.

Things in homology are about measuring defects. Snake lemmas, etc, tell you how those defects are related.

It is just linear algebra, essentially - if something is not injective it has a kernel, etc.

 

[StatusX]

Thanks, that makes sense. In a "perfect" space, with no holes, every cycle would be a boundary, and so the chain groups would form an exact sequence, and the homolgy would be trivial. The homology groups tell us how the space deviates from this, ie, what kinds of holes it has.

But I still don't understand where the actual, perfect exact sequences come into things. For example, take the Mayer Vietoris sequence. To derive this, you form the short exact sequence of chain groups.

0 \rightarrow C_n(A \cap B) \rightarrow C_n(A) \oplus C_n(B) \rightarrow C_n (A+B) \rightarrow 0

Where it can be shown under nice circumstances that the homolgy induced by C_n(A+B) (sums of chains entirely in A or entirely in B) is isomorphic to the homology of their union. Then the snake lemma says this extends to a long exact sequence of homology groups:

...\rightarrow H_{n+1}(A \cup B)\rightarrow H_n(A \cap B) \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n (A \cup B) \rightarrow H_{n-1}(A \cap B) \rightarrow ...

What does this mean? I know how to use it to compute things, and it is very useful for that. But I have no idea what this sequence tells me about the corresponding groups, let alone the corresponding spaces. Like I said, this is a very vauge question, and probably a difficult one to answer. It's probably just something you have to get familiar with, but I'm just looking for a head start in understanding it.

 

[mathwonk]

an exact sequence is just the statement of the rank nullity theorem, but repeated over and over for compositions of maps : i.e. 0-->A-->B-->C-->0 is exact if and only if the kernel of the map B-->C is A, and the map B-->C is onto.

i.e. the rank of the map B-->C is dimC and the nullity is dimA, and these add up to dimB.

the sequence 0-->A-->B-->C-->D-->0 is exact if the kernel of the map B-->C is A, and the quotient of C by the image of the map B-->C is D.

thus dimA-dimB+dimC-dimD = 0, etc...

 

[matt grime]

Mayer Vietoris says that to work out the obstructions in AuB, that's like taking disjoint copies of A and B and looking at their defects, and then counting the ones you counted twice from their intersection that you ignored: it is the inclusion exclusion principle. The snake lemma tells you that if you were looking at n-dim holes, then you next need to take into account the n-1 dim holes - or the boundaries of the n-dim holes.

 

[mathwonk]

do an example, like A,B both discs, and AmeetB
a circle, and AunionB a sphere.presumably knowing the hom ology of a circle should allow you to compute the homology of a sphere.