Re: Nine Lemma
John Baez
John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:
>What's a famous, catchy, exciting or illuminating application
>of the Nine Lemma in homological algebra? The Wikipedia article
>on this subject needs some help, and someone asked me this...
So nobody knows any use for the Nine Lemma? Interesting....
Dimitri Ara
> So nobody knows any use for the Nine Lemma? Interesting....
One can prove the exactness of the relative Mayer-Vietoris sequence
(for singular homology) using the nine lemma.
Let (A_1, A_2) -> (X_1, X_2) be an inclusion of excisive couples. We
have a commutative diagram :
0 0 0
| | |
V V V
0 -> C(A_12) -> C(A_1) (+) C(A_2) -> C(A_1) + C(A_2) -> 0
| | |
V V V
0 -> C(X_12) -> C(X_1) (+) C(X_2) -> C(X_1) + C(X_2) -> 0
| | |
V V V
0 -> C(X_12, A_12) -> C(X_1, A_1) + C(X_2, A_2) -> (C(A_1) + C(A_2))/(C(X_1) + C(X_2)) -> 0
| | |
V V V
0 0 0
where Y_12 = Y_1 \cap Y_2, Y = Y_1 \cup Y_2 and C(Y,B) is the relative
singular chain complex of (Y, B).
The columns and the top rows are exact, so the bottom row is
exact by the nine lemma. I claim that the long exact sequence
associated to this row is the relative Mayer-Vietoris sequence.
To see that, let's consider this morphism :
0 -> C(A_1) + C(A_2) -> C(X_1) + C(X_2) -> (C(A_1) + C(A_2))/(C(X_1) + C(X_2)) -> 0
| | |
V V V
0 -> C(A) -> C(X) -> C(A,X) -> 0
The first arrows are quasi-isomorphisms by hypothesis. So is the third
by the five lemma. QED.
This proof is essentially a reformulation of Spanier's proof.
--
Dimitri Ara
Agusti Roig
It can be used in the proof of the excision theorem for singular
homology (although there are proofs without it).
If you apply it to the commutative diagram with exact rows and columns:
S(A - U) ---> S(A - U) ---> 0
| | |
| | |
V V V
S(A) + S(A - U) ---> S(A) + S(X - U) ---> S(X - U) / S(A - U)
| | |
| | |
V V V
S({A, A - U}) ---> S({A, X - U}) ---> S({A, X - U}) / S({A,
A - U})
you get an isomorphism of complexes
S(X - U) / S(A - U) ---> S({A, X - U}) / S({A, A - U})
which composed with the isomorphism
H( S({A, X - U}) / S({A, A - U})) ---> H(X,A)
gives the excision isomorphism
H(X - U, A - U) ---> H(X,A)
Notations and conventions:
(1) U is a subspace of A , A a subspace of X , with the closure of
U contained in the interior of A .
(2) S(X), S(X,A) stand for the singular and relative chain complexes,
respectively.
(3) S({A, X - U}) is the subcomplex of S(X) generated by those
simplices with their images contained in A or in X - U .
(4) + means direct sum.
(5) H(X), H(X,A) mean singular homology and relative singular
homology, respectively.
Agustí Roig
Mike
An overstatement to say that nobody knows any use for it. To my
knowledge the 9 lemma is never strictly NECESSARY. But it is handy to
have around. It conceptually illuminates homological algebra (assuming
that one has a brain as strange as mine that feels illuminated by the
result). One one has the9 lemma one can give tidy proofs of relative
Mayer-Vietoris and excision as others in the thread have pointed out.
Actually, I have always thought that the hexagonal lemma is
overrated as to usefulness. Opinions everybody?
Mike