Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This is a question about finding references and hopefully a larger context for a lemma in homological algebra I proved recently. The motivation is to understand properties of characteristic classes of $T_f$, the mapping torus of a diffeomorphism $f$ of a closed manifold, by applying the lemma to Mayer-Vietoris and a change-of-coefficients sequence for the cohomology of $T_f$.

Let $C_{ij}, 1 \leq i,j \leq 3$ be cochain complexes, and $$ \begin{matrix} & & 0 & & 0 & & 0 & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & C_{11} & \stackrel{g}\to & C_{21} & \stackrel{h}\to & C_{31} & \to & 0 \\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & \\ 0 & \to & C_{12} & \stackrel{g}\to & C_{22} & \stackrel{h}\to & C_{32} & \to & 0 \\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & \\ 0 & \to & C_{13} & \stackrel{g}\to & C_{23} & \stackrel{h}\to & C_{33} & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 0 & & 0 & & 0 & & \end{matrix}$$

a commuting diagram where the rows and columns are short exact sequences. Let $\delta_H : H^k(C_{3j}) \to H^{k+1}(C_{1j})$ and $\delta_V : H^k(C_{i3}) \to H^{k+1}(C_{i1})$ denote the boundary homomorphisms in the associated long exact sequences. The long exact sequences can be arranged into a commuting grid

$$ \begin{matrix} H^{k-2}(C_{33}) & \stackrel{\delta_H}\to & H^{k-1}(C_{13}) & \stackrel{g}\to & H^{k-1}(C_{23}) & \stackrel{h}\to & H^{k-1}(C_{33}) & \stackrel{\delta_H}\to & H^k(C_{13}) \\ {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ \\ H^{k-1}(C_{31}) & \stackrel{\delta_H}\to & H^k(C_{11}) & \stackrel{g}\to & H^k(C_{21}) & \stackrel{h}\to & H^k(C_{31}) & \stackrel{\delta_H}\to & H^{k+1}(C_{11}) \\ {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ \\ H^{k-1}(C_{32}) & \stackrel{\delta_H}\to & H^k(C_{12}) & \stackrel{g}\to & H^k(C_{22}) & \stackrel{h}\to & H^k(C_{32}) & \stackrel{\delta_H}\to & H^{k+1}(C_{12})\\ {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ \\ H^{k-1}(C_{33}) & \stackrel{\delta_H}\to & H^k(C_{13}) & \stackrel{g}\to & H^k(C_{23}) & \stackrel{h}\to & H^k(C_{33}) & \stackrel{\delta_H}\to & H^{k+1}(C_{13}) \\ {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ \\ H^k(C_{31}) & \stackrel{\delta_H}\to & H^{k+1}(C_{11}) & \stackrel{g}\to & H^{k+1}(C_{21}) & \stackrel{h}\to & H^{k+1}(C_{31}) & \stackrel{\delta_H}\to & H^{k+2}(C_{11}) \\ \end{matrix}$$

The grid is symmetric under translation by 3 steps up and 3 to the right.

Lemma. If $[\alpha] \in H^k(C_{12})$ and $[\beta] \in H^k(C_{21})$ are classes such that $g[\alpha] = u[\beta] \in H^k(C_{22})$ then there is some $[\gamma] \in H^{k-1}(C_{33})$ such that both $\delta_H[\gamma] = v[\alpha] \in H^k(C_{13})$ and $\delta_V[\gamma] = -h[\beta] \in H^k(C_{31})$.

Proof. Take $\chi \in C^{k-1}_{22}$ such that $d\chi = g\alpha - u\beta$. By the definition of the boundary homomorphisms, $d(v\chi) = g(v\alpha)$ implies that $\delta_H([h(v\chi)]) = [v\alpha]$, and $d(h\chi) = -u(h\beta)$ implies that $\delta_V([v(h\chi)]) = -[h\beta]$. Hence we can set $\gamma = vh\chi$.

Does this lemma look familiar? Do you know some place where it's written down?

Edit: Corrected subscripts in statement of lemma.

Update: Thanks for the alternative proofs. However, what I'm after is rather a bibliography reference that I can cite when writing up my application, just to emphasise that it is an instance of something that someone somewhere has already considered (as I imagine it is).

share|improve this question
    
$C_{23}$ should be $C_{13}$ and $C_{32}$ should be $C_{31}$ in the statement of the lemma, no? –  Mariano Suárez-Alvarez Jan 3 '13 at 20:10
1  
Look at your grid as a double complex, and let $C$ be the total complex. There is an action of $G=\mathbb Z$ on it by the translation you described, so we can compute hypercohomology $\mathbb H^\bullet(G,C)$. Using one of the two hypercohomology spectral sequences, we see is zero because of $C$ is exact; the other hypercohomology spectral sequence has then $E_2$ page looking like $H^\bullet(H^\bullet(\mathbb Z,C))$ and converges to zero. Since $\mathbb Z$ has global dimension $1$, this spectral sequence has only two rows (columns?) and degenerates at $E_3$; since the limit is zero, ... –  Mariano Suárez-Alvarez Jan 3 '13 at 20:24
1  
... we really get a lot of isomorphisms. Maybe this is what you are seeing? (I am assuming everything converges; this should follow from the fact that your $C_{i,j}$ are bounded, I think!) –  Mariano Suárez-Alvarez Jan 3 '13 at 20:25
1  
Re: your update: You can refer to this MO question! –  Mariano Suárez-Alvarez Jan 4 '13 at 17:05

3 Answers 3

Everything can be reduced to long exact sequences induced by short exact sequences of complexes.

In your setting, there are short exact sequences of complexes as follows

$$0\rightarrow C_{11}\stackrel{(u,g)}\longrightarrow C_{12}\oplus C_{21}\longrightarrow C_{12}\cup_{C_{11}}C_{21}\rightarrow 0$$

$$0\rightarrow C_{12}\cup_{C_{11}}C_{21}\stackrel{(g,-u)}\longrightarrow C_{22}\stackrel{h\nu}\longrightarrow C_{33}\rightarrow 0$$

This produces long exact sequences

$$\cdots\rightarrow H^{k}C_{11}\longrightarrow H^{k}C_{12}\oplus H^{k}C_{21}\longrightarrow H^{k}(C_{12}\cup_{C_{11}}C_{21})\longrightarrow H^{k+1}C_{11}\rightarrow \cdots$$

$$\cdots\rightarrow H^{k}(C_{12}\cup_{C_{11}}C_{21})\longrightarrow H^{k}C_{22}\longrightarrow H^{k}C_{33}\longrightarrow H^{k+1}(C_{12}\cup_{C_{11}}C_{21})\rightarrow \cdots$$

Your hypotheses say that

$$H^{k}C_{12}\oplus H^{k}C_{21}\longrightarrow H^{k}(C_{12}\cup_{C_{11}}C_{21})\longrightarrow H^{k}C_{22}$$

$$([\alpha],[\beta])\mapsto [\alpha-\beta]\mapsto 0$$

therefore there exists $[\gamma]\in H^{k-1}(C_{33})$ such that

$$H^{k-1}C_{33}\longrightarrow H^{k}(C_{12}\cup_{C_{11}}C_{21})$$

$$[\gamma]\mapsto [\alpha-\beta]$$

Now it is enough to compose with the morphism induced in cohomology by

$$\left(\begin{smallmatrix}\nu&0\\\0&h\end{smallmatrix}\right)\colon C_{12}\cup_{C_{11}}C_{21}\longrightarrow C_{13}\oplus C_{31}$$

in order to obtain the thesis of your lemma. (BTW, notice that there is a misprint in your subscripts, you must replace two 2s by 1s)

share|improve this answer

This has a simple interpretation in terms of spectral sequences. Think of the top left 2x2 square of the original square as a triple complex. Call the 3 dimensions $x$ (horizontal), $y$ (vertical), and $z$ ($C_{ij}$ differential). By using either double complex spectral sequence, we see that the total cohomology of the $xy$-plane is just $C_{33}$. Thus the total cohomology of the triple complex is $H^*(C_{33})$.

On the other hand, we can also compute the total cohomology of the triple complex by a spectral sequence that first takes the $z$-cohomology and then takes the $xy$-cohomology. A pair $([\alpha],[\beta])$ in your lemma gives a class that survives this spectral sequence: $g([\alpha])-u([\beta])$ is the $d_1$ differential, and the $d_2$ differential will vanish for degree reasons. The operation taking $([\alpha],[\beta])$ to $[\gamma]$ is just the isomorphism between the limit of this spectral sequence and the total cohomology $H^*(C_{33})$.

Note that in your proof, $\chi$ is only defined up to a cocycle in $C_{22}$, and so $[\gamma]$ will only be defined modulo the image of $vh:H^{k-1}(C_{22})\to H^{k-1}(C_{33})$. This indeterminacy reflects exactly the fact that $([\alpha],[\beta])$ corresponds to an element of the associated graded of a filtration on $H^{k-1}(C_{33})$ (whose first term is the image of $vh$), rather than an element of $H^{k-1}(C_{33})$ itself.

share|improve this answer

One application of your lemma is in differential cohomology. See e.g. Ex. 3.25 in arxiv.

I would be very interested in a generalization of this lemma to triangulated categories. So replace your grid of exact sequences by a grid of triangles in a triangulated category. Instead of cohomology you consider the group Hom(T,...) for a fixed object T. Then you get similar long exact sequence and can state an analogous lemma. Is there a proof in this generality?

share|improve this answer
2  
Fernando's proof works for any homological functor on a triangulated category. –  Eric Wofsey Jan 3 '13 at 21:39
    
Thanks. The claim in the example from your paper looks slightly different to me, but this kind of reference is helpful. –  Johannes Nordström Jan 3 '13 at 23:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.