A lost lemma about periodicity in a grid of long exact sequences?

Question

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).

Answer 1

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.

Answer 2

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)

Answer 3

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?

Answer 4

Versions of this result that work in other (closed symmetric monoidal) triangulated categories can be found in Bruner-Greenlees (Experiment. Math., 1995, Lem. 2.2), Andrews-Miller (Journal of Topology, 2017, Lem. 9.3.2) and Bruner-Rognes (Transactions of the AMS, 2022, Prop. 2.3).