Hi all,

Today I was playing with the cohomology of manifolds and I noticed something intriguing. Although I might just have been caught out by a couple of enticing coincidences, it feels enough like there might be something going on that I thought I'd put it out here.

Let $M$ be an $n$-manifold with boundary $\partial M$. We write out the long exact homology sequence for the pair $(M, \partial M)$:

$$\cdots \to H_k(M) \to H_k(M, \partial M) \to H_{k - 1}(\partial M) \to \cdots$$

Let's apply Poincaré duality termwise, and keep the arrows where they were out of sheer faith. What we get is

$$\cdots \to H^{n - k}(M, \partial M) \to H^{n - k}(M) \to H^{n - k}(\partial M) \to \cdots$$

Surprisingly, this is the long exact *cohomology* sequence for the pair $(M, \partial M)$! To my mind, two things here are weird. The first is that intuitively, any functoriality properties Poincaré duality possesses should be arrow-reversing. The second is that we have a shift - but not a shift by a multiple of 3. So the boundary map in the homology sequence 'maps' to something that doesn't change degree in the cohomology sequence.

Let's play the same game with the Mayer-Vietoris sequence. For simplicity, suppose now $M$ is without boundary. Write $M = A \cup B$ where $A$ and $B$ are $n$-submanifolds-with-boundary of $M$ and $A \cap B$ is a submanifold of $M$ with boundary $\partial A \cup \partial B$. Then we have

$$\cdots \to H_k(A \cap B) \to H_k(A) \oplus H_k(B) \to H_k(M) \cdots$$

Hitting it termwise with Poincaré duality, and cruelly and unnaturally keeping the arrows where they are once again, we get

$$\cdots \to H^{n - k}(A \cap B, \partial A \cup \partial B) \to H^{n - k}(A, \partial A) \oplus H^{n - k}(B, \partial B) \to H^{n - k}(M) \to \cdots$$

This looks unfamiliar, but by looking at cochains it's not hard to see that there actually is a long exact sequence with these terms. However, this time we don't have the weird shift.

Now is there anything going on here, or just happenstance? Is there really a sense in which Poincaré duality is functorial with respect to long exact sequences? If so, what's the 'Poincaré dual' of the long exact sequence of the pair $(M, N)$ where $N$ is a tamely embedded submanifold of $M$?

Edit: Realised that in the final l.e.s. the arrows should actually go the other way, which is slightly less impressive. Even so...