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matt grime

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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.

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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.

[tex]0 \rightarrow C_n(A \cap B) \rightarrow C_n(A) \oplus C_n(B) \rightarrow C_n (A+B) \rightarrow 0[/tex]

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:

[tex]...\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 ... [/tex]

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.

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mathwonk

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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.....

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matt grime

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- #6

mathwonk

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a circle, and AunionB a sphere.

presumably knowing the hom ology of a circle should allow you to compute the homology of a sphere.

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